yf^\ 


^ 


7  C^^l'^cZt^f^^^<i--~^=a,^y~iS'^^^ 


f^^■^^«■A^-^ 


ELEMENTS 


OP 


ANALYTICAL  GEOMETRY. 


ELEMENTS 


Of 


ANALYTICAL  GEOMETRY 


AND      OF      THE 


DIFFERENTIAL  AND  INTEGRAL 


CALCULUS. 


CHARLES    DAVIES,    LL.D., 

PEOrESSOB  OF  HI6HEE  MATHEMATICS,  COLTTMBIA  COLLBOI. 


NEW    YOEK: 
PUBLISHED   BY  A.   S.   BARNES   &   Co., 
Ill  &  113  WILLIAM   ST.,  cor.  JOHN. 

1866. 


jsiTTTEEED,  according  to  Act  of  Congresfl,  In  the  year  Eighteen  Hundred  and  Sixty, 

Bt     CHARLES     DAVIES, 

In  the  Clerk's  Office  of  the  District  Court  for  the  Southern  District  of  New  York. 


William  Dentse,  6.  "W.  "Wood, 

StereoUjper  and  Electrotyper^  Printer^ 

183  William  St.,  New  York.  OoT.  Dutch  &  John  8t» 


SSI 
dl8h£. 


PREFACE. 


The  method,  invented  by  Descartes,  of  represeutmg  all 
the  parts  of  a  geometrical  magnitude  by  a  smgle  equation, 
has  wrought  an  entire  change  in  the  mathematical  and 
physical  sciences. 

Before  his  time,  the  Measures  of  the  Geometrical  Mag- 
nitudes were  alone  subjected  to  the  processes  of  Algebra. 
He  extended  the  analysis  to  the  determination  of  their 
positions  and  forms.  By  this  happy  invention,  modes  of 
investigation  at  once  difficult  and  disconnected,  and  de- 
pendent for  success,  in  each  particular  case,  on  the  skill 
and  ingenuity  of  the  inquirer,  and  often  on  accident,  are 
reduced  to  a  simple  and  uniform  process.  The  great 
work  of  La  Place,  is  the  legitimate  fruit  of  this  discov- 
ery. Here,  all  the  formulas  necessary  for  determining 
the  positions  and  motions  of  the  bodies  of  the  solar  sys- 
tem, are  deduced  from  the  single  law  of  gravitation,  and 
expressed  in  general   equations. 

The  first  edition  of  the  Analytical  Geometry  was  pub- 
lished in  1836.  It  was  designed,  specially,  for  the  pupils 
of  the    Military  Academy,    and    in  its    construction,   little 


Vi  PREFACE. 

attention  was  paid  to  the  supposed  wants  of  other  In 
stitutions.  It  is  now  presented  to  the  public  under  a 
different  form,  and  designed  to  fill  a  very  different  place 
In  1836,  the  study  of  Mathematics,  and  especially  in  the 
higher  branches,  was  limited,  comparatively,  to  a  few  In- 
stitutions of  the  higher  grade.  Now,  it  is  pursued  in  all 
our  Colleges,  and  in  many  of  our  Academies  and  High 
Schools.  It  forms  a  part,  and  indeed  an  important  part, 
of  our   system   of   PubUc  Instruction. 

To  prepare  a  work  that  shall  cultivate  this  growing 
taste  for  mathematical  science,  in  one  of  its  most  attract- 
ive departments — not  too  large  for  general  use,  and  yet 
containing  all  the  great  principles,  rightly  arranged  and 
properly  discussed  —  has  been  attended  with  many  diffi- 
culties. How  far  they  have  been  overcome,  the  public 
will  judge.  The  present  work  is  supposed  to  contain  all 
that  is  necessary  to  the  general  student.  The  Table  of 
Contents  indicates  the  subjects,  and  the  order  in  which 
they  are  treated. 

Columbia  College,  New  York^  1860. 


CONTENTS. 


INTRODUCTION. 

PAOB. 

Geometrical  Magnitudes 18 

Construction  of  Algebraic  Expressions 13-20 

Construction  of  the  roots  in  the  First  form 'iO 

"                         «                 Second  form 21 

"                         "                 Third  form , 22 

"                         "                Fourth  form...'. 23 


BOOK  I. 

POINTS       AND       LINES       IN       A       PLANE. 

Analytical  Geometry  defined 26 

Points  in  the  different  angles 26-29 

Examples  in  their  construction 29-S(» 

Co-ordiuate  Axes — Origin 30 

Straight  Line  in  a  Plane 81 

Equation  of  a  Straiglit  Line 31  -"6 

Construction  of  Straight  Lines. ' 36— i() 

Equation  of  the  First  Degree  between  two  Variables. 40 

Distance  betw^een  two  Pc'nts 41 

Equation  of  a  Line  passing  through  a  given  Point 42 

Equation  of  a  Line  passing  through  two  given  Points 43 

Equation  of  a  Line  parallel  to  a  given  Line 45 

Anple  included  between  two  Lines 46 

Condition  of  two  Lines  intersectin?  each  other 47 

A  Perpendicular  to  a  given  Line,  from  a  given  Point 48 


nil  CONTENTS. 

PASK. 
TKAN3FJRMATI0N       OF      CO-ORDINATES. 

Formulas  fc?  passing  from  one  System  of  Axes  to  a  Parallel  System  52 

To  pass  from  a  Rectangular  to  an  Oblique  System 53 

To  pass  from  Rectangular  to  Rectanglular 64 

To  pass  from  an  Oblique  to  Rectangular 54 

Remarks 55-57 

rOLAB        CO-ORDINATES. 

Formulas 5*7-59 

BOOK  II. 

or       THE         CIRCLE. 

Equation  of  a  Line  defined 60 

Equation  of  the  Circle — Origin  at  the  centre 60 

Interpretation  of  Equation 61-65 

General  Form  of  Equation 65 

Supplementary  Chords 66-68 

Tangent  Line  to  the  Circle 68-7 1 

Normal  Line *. 71-72 

Polar  Equation 72-73 

Interpretation  of  Polar  Equation 73-75 

BOOK  m. 

OF       THJS       ELLIPSE. 

Ellipse  defined ,  76 

Construction  of  the  Ellipse 76-70 

Equation  of  the  Ellipse 79-81 

Interpretation  of  the  Equation 81-84 

Eccentricity 84 

Polar  Equation 84-86 

Diameters 86 

Every  Diameter  bisected  at  the  Centre 86-87 

Ordinates  to  Diameters 87-83 

Parameter ,  88-89 


CONTENTS.  IX 

PAGB, 

Ellipse  and  Circumscribing  Circle 89-90 

Ellipse  and  Inscribed  Circle 91 

Equation  of  Tangent 92-94 

Normal 9-1-06 

Tangent  line  and  Lines  drawn  to  Foci 96 

Supplementary  Chords 9*7-99 

Supplementary  Chords,  Tangent,  and  Diameter 99 

Construction  of  Tangent  Lines  to  an  Ellipse 100-103 

ELLIPSE   KEFEKKED   TO   CONJUGATE   DIAMETEKS. 

Definition  of  Conjugate  Diamctrrs '. 103 

Equation  of  the  Ellipse  referred  to  Conjugate  Diameters 105-106 

Relation  of  Ordinates  to  each  other 106-107 

Parameter 107 

Relation  between  the  Axes  and  Conjugate  Diameters , 107-109 

L-iterprctation  of  the  Equations 1 09-1 1 1 

BOOK   IV. 

OF        THE        TAR  A  B  O  L  A. 

Parabola  defined 112 

Equation  of  the  Parabola US 

Interpretation  of  the  Equation 114 

Parameter 115 

Relation  of  the  Ordinates  and  Abscissas 116 

Polar  Equation 117 

Interpretation  of  the  Polar  Equation 118 

Tangent  to  the  Parabola 119 

^ub-tangent 120 

Normal  and  Sub-normal 121 

Perpendicular  from  Focus  to  Tangent ...  122 

Construction  of  Tangent  Lines  to  the  Curve 123-1 25 

Equation  of  the  Parabola  when  referred  to  Oblique  Axes 126 

Interpretation  of  the  Equation 127-129 

1* 


CONTENTS. 


BOOK  y. 

OF   THE   HYPERBOLA   AND   ALOEBBAIO   CURVES. 

Hyperbola  defined 130 

Coustruction  of  the  Curve 130-133 

Equation  of  the  Curve 133-136 

Interpretation  of  the  Equation 135-13fi 

Eccentricity 13s 

Polar  Equation 1 38-14("» 

Diameters — Their  Properties , 140-141 

Paraniefer : 141 

Equation  of  the  Tangent — Sub-tansjent 142 

Equation  of  the  Normal — Sub-normal 143 

Tangent  bisects  Ande  of  Linos  to  Foci 144 

Suppl»^meutarv  Chords 145 

Construction  of  Tancent  Lines 146 

Conjugate  Diameters   147 

Equation  when  referred  to  Conjugate  Diameters 148 

Interpretation  of  the  Equation 150 

Relation  between  Axes  and  Conjugate  Diameters 151-152 

Hyperbola  referred  to  its  Asymptotes 152-154 

Equation  of  the  Curre 154 

Interpretation  of  the  Equation   155 

Asvmptotes  approach  the  Curve 156 

Asymptotes,  the  Limits  of  Tangents 166 

ALGEBRAIC       CURVES. 

Algebraic  <^urves  defined 15? 

Equation  of  the  Second  Degree  between  Two  Variables 158 

Change  in  the  direction  of  the  Axes 158 

Change  of  the  Origin  of  Co-ordinates 1  <*'0 

Interpretation  of  the  Equations 16«>-1  ^^2 

Classification  of  Lines ^^'^ 

Equation  when  the  Origin  is  in  the  Curve 163 


CONTENTS.  Xi 


BOOK  VI. 

SPACE  —  POINT   AND   LINE  — PLANE  —  SUBJACES. 

Space  defined 165 

Co-ordinate  Planes — Axes — Angles 165-171 

Distance  between  two  Points 171 

Line  and  Co-ordinate  Axes 172 

Equations  of  a  Straight  Line  in  Space 173-176 

Interpretation  of  the  Equations 175-177 

Equations  of  a  Line  passing  through  two  Points 177-179 

Lines,  Intersecting  and  Parallel 179-lbl 

Angle  between  two  Lines 181-185 

Examples  in  Construction 185-186 

OF      THE      PLANK. 

Equation  of  a  Plane  defined 186 

Equation  of  a  Plane 186-188 

Traces  of  a  Plane 188-190 

Line  Perpendicular  to  a  Plane 190-191 

SURFACES       OF      THE      SECOND       ORDER. 

Equation  of  a  Surface  defined 192-194 

Surfaces  of  Revolution 194 

Equation  of  Surfaces  of  Revolution 196 

Sphere — EUipsoid — Paraboloid — Hyperboloid 196-198 

Surfaces  of  Single  Curvature 198 

Equation  of  the  Surface  of  the  Cylinder 199 

Equation  of  the  Surface  of  the  Cone 199-200 

Intersectionofa  Conic  Surface  by  a  Plane 201 

Circle— Ellipse— Parabola— Hyperbola 202-204 


ANALYTICAL    GEOMETRL 


INTKODUCTION. 

1.  There  are  four  kinds  of  Geometrical  Magnitude :  viz. 
Lines,  Surfaces,  Volumes,  and  Angles.  In  Geometry  these 
magnitudes  are  presented  to  the  mind  by  a  pictorial 
language,  and  their  properties  are  deduced  by  constant 
reference  to  the  magnitudes  themselves. 

Instead,  however,  of  keeping  the  magnitudes  constantly 
before  the  mind,  we  may,  if  we  please,  denote  them  by 
Algebraic  symbols.  Having  done  this,  we  operate  on  these 
symbols  by  the  known  methods  of  Algebra.  The  results 
obtained,  will  be  equally  true  for  the  geometrical  magnitudes 
and  the  symbols  by  which  they  are  represented.  We  next 
interpret  these  results  ;  and  this  leads  to  the  development 
of  the  properties  of  Geometrical  Magnitudes. 

GEOMETRICAL      CONSTRUCTIONS. 

'  2.  The  coNSTBUcnoN  of  an  Algebraic  expression,  is  the 
operation  of  finding  a  geometrical  magnitude  equivalent 
to  it. 

Construction  of  expressions  of  the  first  degree. 

3.     Construct  the  expression,     a  ■+■  b. 


14  ANALYTICAL      GEOilETEY. 

Draw   an    indefinite    right  line 

AB.     From  any  point  as  A^  lay  \ • \—B 

off  a  distance  AG  equal   to    «, 

and  then  from  (7,  a  distance   CD  equal  to  J,  and  AD  will 

be  equivalent  to    a  -\-  h. 

4.  Construct  the  expression,     a  —  h. 
Draw    an   indefinite    right   line 

AB.     From  any  pomt  as  ^,  lay  ^  t;  ~t7~'^ 

off  a  distance  AD  equal    to    a, 

and  then  from  jP,  a  distance  DC,  in  the  direction  toicards 
A,  equal  to  5 ;  AC  will  then  express  the  difference  between 
a  and  J,  and  hence,  is  equivalent  to     a  —  h. 

If  5  is  greater  than  a,     a  —  h 

wdll  be  essentially  negative  :  A  C  i \ 1—  ^ 

will  then   be  negative,   which  is 

shown  by  the  point  C  falling  at  the  left  of  A. 

We  have,  in  this  example,  the  geometrical  interpretation 
of  the  negative   sign:   viz. 

If  distances  in  one  direction  are  regarded  as  2^ositivey 
those  in  a  contrary/  direction  must  be  regarded  as  nega- 
tive* 

5.  Construct  the   expression,     a  —  b  ■}-  c  —  d. 

Draw    an    indefinite    line  AB, 
From    any  point,  as    ^4,   lay   off       Jr    j^     "^     ^     ^i    ^ 
the  distance  AC  equal  to  a,  and 
then  the   distance   CD^  in  the   opposite   direction,  equal  to 

*  Bourdon,  Art.  89.  University,  Art.  96. 

Note. — All  the  references  are  to  Dayies'  Bourdon,  Davies'  University 
Algebra,  and  Davies'  Legendre,  Geometry,  and  Analytical  Trigonometry. 


INTRODUCTION. 


15 


b.  From  Z>,  lay  off  BE^  to  the  right,  equal  to  o,  and 
fthen  from  jEJ  lay  off  JEF^  to  the  left,  equal  to  d.  The 
line  AF  will  be  equivalent  to  the  algebraic  expression. 
It  will  be  negative,  because  the  sum  of  the  negative  terras, 
in  the  algebraic  expression,  exceeds  the  sum  of  the  posi- 
tive terms,  and  this  is  indicated  by  the  direction  of  the 
line  AF     From  the  above  examples,  we  conclude  that, 

Every  algebraic  expression  of  the  first  degree  will  repre- 
sent a  line;  whence,  it  is  called,  linear, 

6.     Construct  the  expression,    — 

Draw  two  indefinite  right  lines  AJB^ 
AF,  making  any  angle  with  each 
other.  From  A,  lay  off  a  distance 
AG—Cj  also  the  distance  AJ3  =  a. 
Then  from  A,  lay  off  AD  =  b; 
join  C  and  Z>,  and  through  J]  draw 
JBF  parallel  to  CD;  then  will  AF  be  equivalent  to  the 
given  expression.    For,  we  have  by  similar  triangles,* 

AC 


that  is, 
therefore, 


AB  :  :  AD  :  AF; 
a     ::      b    :  AF; 


AF  = 


ab 


^  Construction  of  expressions  of  the  second  degree. 

7,     Construct  the   expression,     ab. 

The  degree  of  a  term  is  the  number  of  its  literal  factors.f 
Hence,   ab    is  of  the  second  degree. 


*  Legendre,  Bu.  IV.     Pi  op.  15.         j    bourdou,  Art.  25.    Univ.  Aj-t.  12. 


16 


ANALYTICAL      GEOMETKY 


B  B 


Draw  the  indefinite  straight  line 
AJB.  Lay  off,  from  A  to  Z>,  as  many 
units  of  length  as  there  are  units  in 
a.  At  X>,  draw  DC  perpendicular 
to  AB^  and  make  it  equal  to  as  many 

units  of  length  as  there  are  units  in  h.  Then,  the  rect- 
angle ADCE^  will  contain  as  many  units  of  surface  as 
there  are  units  in  the  expression,  a  X  5.  Hence,  we  con- 
clude  that, 

Every   algebraic  expression  of  the  second   degree    repre- 
sents a  surface:^ 


I 


Construction  of  an  expression  of  the  third  degree. 

8.     Construct  the   expression    ahc. 

Draw  an  indefinite  line,  and  lay  off 
AB  equal  to  the  number  of  units 
in  a.  Draw,  in  the  plane  of  the  paper, 
jB(7,  perpendicular  to  AB^  and  make 
it  equal  to  h.  At  (7,  suppose  CD  to 
be    drawn    perpendicular   to    the   plane  A      a     B 

of   the    paper,    and   made   equal    to    c. 

Then,  having  drawn  the  other  lines  of  the  figure,  ABC-D 
will  be  a  rectangular  parallelopipedon  equivalent  to  the 
expression   ahc.     Hence, 

Every  algebraic  expression  of  the  third  degree  will  repre- 
sent a  volume.] 


Construction  of  expressions  of  the  zero  degree. 
a 


9.     Construct  the  expression. 


b 


*  Logendre,  Bk.  IV.     Prop.  4.     Sch.  1.         f  Bk.  VII.  Prop.  13.  Cor. 


INTRODUCTION. 


IT 


Since  there  is  one  literal  factor  in  the  numerator,  and 
one  in  the  denominator,  the  quotient 
is  an  abstract  number;  and  hence,  con- 
tains no  literal  factor:  therefore,  the 
degree  of  the  term  is  0.  Draw  AH^ 
and  make  it  equal  to  h.  At  J9^  draw 
JZ5,  perpendicular  to  AH^  and  make 
it  equal  to  a,  and  join  A  and  B.    Then, 

-  =  tan  A  to  the  radius  1.* 
o 

If   a    were    made    the    hT]^othenn«!o, 

-  would  denote  the  cosine  (7,  or  sin  i? 
a 

to  the  radius  1.     Hence, 

Every  algebraic  expression  of  the  zero  degree  represents 
the  s,ine^  cosine^  tangent^  cfic,  of  an  arc  or  angle^  to  the 
radius  1. 

It  follows,  therefore,  that  every  abstract  number  has  a 
geometrical  interpretation;  for,  it  will  always  denote  some 
function  of  an  arc  described  with  the  radius  1. 


Homogeneity  of  terms. 

10.  We  have  seen,  that  there  are  four  kinds  of  alge- 
braic terms,  wliich  may  represent  geometrical  magnitudes;-. 
viz.  terms  of  the  1st  degree,  which  represent  lines;  terms 
of  the  2d  degree,  which  represent  surfaces;  terms  of  the 
third  degree,  which  represent  volumes ;  and  terms  of  the 
zero  degree,  which  represent  the  functions  of  angles  to  the 
radius  1. 

Since  no  other  magnitudes   occur  in   geometry,  no  alge- 
*  Legendre,  Trig.  Art.  37. 


18  ANALYTICAL      GEOMETRY. 

braic  term  of  a  higher  degree  than  the  third,  can  have  a 
geometrical  equivalent.  K  such  a  term  occur,  we  can  only 
find  its  geometrical  equivalent  by  regarding  all  the  factors 
but  three,  as  numerical. 

Since  like  quantities,  only,  can  be  added  or  subtracted, 
it  follows,  that  if  two  or  more  terms  are  connected  to- 
gether by  the  signs  +  or  —  ,  they  must  be  homogeneous,* 
If  they  are  not  so,  in  form,  it  is  because  the  geometrical 
unit  of  length,  generally  denoted  by  1,  has  been  omitted 
in  the  algebraic  expressions,  wherever  it  has  occurred  as 
a  factor  or  a  divisor;  and  this  must  be  restored  before 
finding  the  geometrical  equivalent.    Thus,  if  we  have, 

ah  +  c, 

the  first  term  is  of  the  second  degree,  and  the  second, 
of  the  first  degree.  The  degree  of  the  second  term  is 
changed  (without  altering  its  numerical  value),  by  intro- 
ducing the  linear  unit  1,  as  a  factor,  and  we  then  have, 

ah  +  \  X  c^ 

vhich  is  a  homogeneous  expression. 
If  we  have  the  expression, 

a  +  he  —  dfg, 

it  may  be  made  homogeneous  by  introducing  the  factors 
of  1 ;  we  then  have, 

Ixlxa+lxJc  —  dfg\ 

which  is  homogeneous. 

•  Bourdon,  Art.  26.     University,  Art.  IS*, 


INTRODUCTION. 


19 


EXAMPLES  IN  CONSTEUCTION. 

1.  Construct  the  expression,    a^  +  li^. 

Draw  an  indefinite  right  line, 
and  lay  off  -4  (7  =  5.  At  (7,  draw 
CB  perpendicular  to  AG,  and 
make  it  equal  to  a,  and  draw  AB : 

2 

then,  AB  will  be  the  equivalent  of 


2.  Construct  the  expression,    c^  —  h"^. 

Draw  an  indefinite  line,  and  on 
it  lay  off  AB  =  c;  and  on  AB^ 
as  a  diameter,  describe  a  semicir- 
cumference  ACB,  With  -4  as  a 
centre,  and  a  radius  equal  to  b, 
describe  an  arc,  intersecting  the  circumference  at  C    Then 

draw  AG  and  GB. 

2         2 

Now,  since  A  GB  is  a  right-angled  triangle,  AB  —AG 
— 2  — 2 

is  equivalent  to    GB  ;f  hence,     GB     is  the   equivalent  of 

c2  -  52. 

3.  Construct  the  expression,    -y/ab,    and    y^. 

Draw  an  indefinite  right  lino 
ABGj  and  from  any  point,  as  Aj 
make  AB  =  a,  and  then  BG  =  b. 
On  ^  (7,  as  a  diameter,  describe  a 
eemi-circumference,   and    from    J?, 

draw  ^Z>  perpendicular  to  AG,  intersecting  the  circumfer- 
ence at  1) :  then,  BB  will  be  the  equivalent  magnitude. 
'       •  Leg.,  Bk.  IV.    Prop.  11.        f  Leg.,  Bk.  IV.    Prop.  11,  Cor.  1. 


20  AXALTTICAL      GEOilETET. 

For,  BD^  =  AB  X  BC"  =  a  x  b; 

hence,  by  extracting  the  square  root  of  both  members, 
J3D  =  yoJ. 

If  we  have   -)/«,  we  have  simply  to  introduce,  under  the 

radical,  the  factor  1,  thus  making  the  expression  of  the 
second  degree. 

"We  then  have,  yo"  =   -/a  x  1. 

Making  AB  =  a,  and  BG  =  1^  we  have  the  same  con- 
struction as  before. 

4.     Construct  the  roots  of  the  equation  of  the  first  form, 

£c2  -f  2px  =  q.\ 

After  making  the  second  member  of  the  equation  homo- 
geneous, and  placing  it   equal  to    ^^,    we  have, 

x^  4-  ^jyx  =  I  X  q  =  b\ 

This  equation   can  be  put  under  the  form, 

x{x  +■  2/?)   =  52 . 

from  which  we  see,  that  5  is  a  mean  proportional  between 
X  and  X  +  2p. 

To  construct  these  values 
of  X,  draw  AB,  and  make  it 
equal  to  b.  At  B,  erect  the 
perpendicular  BC,  and  make 
it  equal  to  j9,  and  join  -4 
and  C.  With  C  as  a  centre, 
and  CB  as  a  radius,  describe 
*  Leg.,  Bk.  lY.   Prop.  23,  Cor.  2.      f  Bour.,  Art  117.      Univ.,  Art  U7. 


INTRODUCTION.  21 

a  semi-circumference  cutting  AC  ia  ^,   and  AG  produced, 
in  J)  I   then  will  A£J  be   equal  to  x.    For,* 

AJE:{AB+  2EG)  =  Xe'  =  b^; 
or,  x{x  -f  2p)  =   b\ 

Finding  the  roots  of  the  given  equation,  we  have, 

jc'  =  —p  +  V&M^S    and    jb"  =   —p  —  y/b^lTj^, 

Having  constructed  the  triangle  AB  C,   as   before,  A  0  will 
represent  the  radical  part  of  the  values  of  x. 

For  the  first  value  of  x,  the  radical  is  positive,  and  is 
laid  off  from  A  towards  C:  then  —  ^  is  laid  off  from 
C  to  JBIy  leaving  AJEJ  positive,  as  it  should  be,  since  it 
is  estimated  from  A  towards  C. 

For  the  second  value  of  a?,  we  begin  at  D,  and  lay 
off  DC  equal  to  —  j9 ;  we  then  lay  off  the  minus  radi- 
cal from  G  to  A^  giving  —  DA,  for  the  second  value 
of  X. 

Let  us  now  see  if  this  value  of  x  will  satisfy  the  equa- 
tion, 

—  x{-  X  +  2p)  =  ^.2, 

or,  -  AD(-  AE)  =  l\ 


or,  AD  X  AE  =  AB  . 

Hence,  the  two  values  of  x,   are    +  AE,   estimated  from 
'^A  towards  D,  and  —  DA,  estimated  from  D  towards  A. 

5.     Construct  the  roots   of  the   equation   of  the  second 
form, 

a^  _  2px  =  q  =  1  X  q  =  b^, 

*  Legendre,  Bk.  IV.     Prop.  30. 


22 


ANALYTICAL      GEOMETEY. 


Finding  the  roots  of  tliis  equation,   we  have, 

and    x' 


=  JO  —  V^^  +  J»^. 
Having  constructed  the 


x'  =  'p-\-  ^/IP-  +  x>\ 

To  construct  these  values  of  x. 
figure,   as    in   the   last    exam- 
ple,  the  first  value   of  x  will 
be    the    line    AD^    estimated 
from  A  to  D. 

The  second  value  will  be 
4-  EC  —  CA,  the  first  esti- 
mated from  E  to  (7,  and  the 
latter  from  C  to  ^ :  this  leaves, 
for  the  reduced  value    —  EA,   estimated  from  E  to  A. 

The  positive  root,  in  the  construction  for  the  first  form, 
corresponds  to  the  negative  root  in  the  construction  for 
the  second;  and  the  negative  root  in  the  first,  to  the 
positive  root  in  the  second.  This  is  as  it  should  be,  since 
either  of  the  forms  changes  to  the  other,  by  substituting 
—  X    for    X. 

6.     Construct    the    roots   of  the    equation    of-  the    third 

fornL 

aj2  +  <2px  =   —  q  =  — 1X2'=   "-^^' 

Solving  the  equation,  we  have, 

x'  z=  —p  +  vi>2  _  52^     and    x"  z=  —  p  —  ^/^-^, 

To  construct  these  values,  draw  an  indefinite  right  line 
FA^  and  from  any  point,  as 
A^  lay  off"  a  distance  AD  =  — jo, 
and,  since  p  is  negative,  we  lay 
off  its  value  to  the  left.  At 
J9,  draw  DC  perpendicular  to 


INTEODUCTION, 


23 


"FA  and  make  it .  equal  to  h.  With  (7  as  a  centre,  and 
CB  —  p  as  a  radius,  describe  the  arc  of  a  circle  cutting 
FA^  in  B  and  E.  Now,  the  value  of  the  radical  quantity 
will  be  BD  or  DE.  The  first  value  of  x  ^vill  be  -  AB 
plus  DE^  equal  to  —  AE.  The  second,  will  be  —  AB  + 
( —  BB^  equal  to  —  AB :  so  that  both  of  the  roots,  being 
negative,  are  estimated  in  the  same  direction  fi-om  -4,  to 
the  left. 
Therefore,  the  two  roots  are    —  AE^    and     —  AB. 


*I,     Construct    the  roots  of  the   equation    of  the  fourth 
form, 

a;2  —  2jtxc  =   —  q  =   —  1  x  q  =   —  b\ 

Solving  the  equation,  we  have. 


p  +  V^2  _  j2^     and    x"  =  p  —  Vp2 


b^. 


A    ^    --....D_^,'-B 


To  construct  these  values  of  x.     Construct  the  radical  part 
of  the  values  of  x,  as  in  the 
last    case.      Then,   since   p    is  ^ 

positive,  we  lay  off  its  value 
AB^  from  A  towards  the 
right.  To  AB,  we  add  BB, 
which  gives  AB,  for  the  first 

value  of  X,  K  from  AB,  we  subtract  BE,  the  remainder, 
AE,  is  the  second  value  of  x.  Both  values  are  positive, 
and  are  estimated  in  the  same  direction,  from  A  to  the 
right. 

In  the   last   two   forms,   if  p   and   h   are   equal,    the   two 
ralues  of  x  become  equal  to  each  other.* 

The    geometrical    construction    conforms    to    this    result. 


*  Bourdon,  Art.  11*3-117.     University,  Art.  146. 


24 


ANALYTICAL      GEOMETRY. 


For,    when     p  =  b,     the   arc  of  the  circle  described  with 

the  centre   C\  will  be  tangent 

to   AJ^,   at  JD'j   and  the  two 

points   ^   and    B   will    unite, 

and  each  root  will  become  equal 

to  AD. 

If  P  is  greater  than  p^,  the 
value  of  jc,  in  the  last  two  forms,  will  be  imaginary.* 

The  geometrical  construction  also  indicates  this  result. 
For,  if  b  exceeds  />,  the  .  circle  described  with  the  centre 
(7,  and  radius  equal  to  />,  will  not  cut  the  line  AB, 
Hence, 

T/ie  imaginary  roots  of  an  equation  give  rise  to  con- 
ditions in  the  construction  which  cannot  be  fulfilled/  and 
t/iis  should  be  so,  since  the  imaginary  roots  can  never 
appear,  unless  the  conditions  of  the  equation  are  incon- 
sistent with  each  other. 


Bourdon,  Art.  116-117.    University,  Art  U6. 


ANALYTICAL    GEOMETRY. 


BOOK    I. 

POINT     AND      STRAIGHT     LINE PROBLEMS TRANSFORMATION 

OE     CO-ORDINATES POLAR     CO-ORDINATES. 

Definitions. 

1.  Analytical  Geometry  is  that  branch  of  Mathematics 
which  has  for  its  object  the  determination  of  the  forms  and 
properties  of  the  Geometrical  Magnitudes,  by  means  of 
Analysis. 

2.  In  Analytical  Geometry,  the  quantities  considered  may 
be  divided  into  two  classes : 

1st.  Constant  quantities^  which  preserve  the  same  values 
in  the  same  investigation;  and, 

2d.  Variable  quantities,  which  assume  all  possible  values 
that  will  satisfy  the  equation  which  expresses  the  relation 
between  them. 

The  constants  are  denoted  by  the  first  letters  of  the 
"alphabet,  a,  b,  c,  &c.;  and  the  variables,  by  the  final  let- 
ters, X,  y,  z,  &c. 

3.  The  terms,  straight  line,  and  plane,  are  used  in  their 
most  extensive  signification. 

That  is,  the  straight  line  is  supposed  to   be  indefinitely 
2 


26  ANALYTICAL       GEOMETRY.  [bOOK   I. 

prolonged,  in  both  directions  ;  and  the  plane,  to  be  indefi- 
nitely extended. 

Points  in  the  same   plane. 

4.  We  shall  first  explain  the  manner  of  determining,  by 
the  algebraic  symbols,  the  position  of  points  in  a  given 
place. 

For  this  purpose,  draw,  in  the 
plane,  any  two  lines,  as  X'AJl, 
TAT\  intersecting  at  A,  and 
making  with  each  other  a  given 
angle,   TAX. 

The  line  XX,  is  called  the 
axis  of  abscissas,  or  the  axis  of 

X;  and  YY^,  the  axis  of  ordi?iates,  or  the  axis  of  Y,  The 
two  taken  together,  are  called  the  co-ordifiate  axes ;  and  the 
point  A,  where  they  intersect,  is  called  the  origin  of  co- 
ordinates. 'The  angle  YAX  is  called,  the  first  angle; 
YAX\  the  second  angle  ;  XA  Y^  the  third  angle ;  and 
Y'AX^  the  fourth  angle. 

First  Angle. 

5.  Let  P  be  any  pomt  in  the  given  plane.  Through  P, 
draw  PD,  parallel  to  AY^    and 

PC,  parallel  to  AX.  Then,  AD, 

or  CP,  is  called  the  abscissa  of  /- — -, 

the  point  P  ;  PD,  or  A  C,  is 
called  the  ordinate  of  P;  and 
the  lines  PP,  PC,  taken  to- 
gether, are  called  the  co-ordi- 
nates of  the  point  P. 


J 


BOOK   I.]        POINT      AND      STRAIGHT      LINE.  27 

Hence,  the  abscissa  of  any  point,  is  its  distance  from  the 
axis  of  ordinates,  measured  on  a  line  parallel  to  the  axis 
of  abscissas;  and  the  ordinate  of  any  point,  is  its  distance 
from  the  axis  of  abscissas,  measured  on  a  line  parallel  to 
the  axis  of  ordinates.  The  co-ordinates  may  also  be  meas- 
ured on  the  axes  themselves.  For,  AD,  AC,  are  equal  to 
the  co-ordinates  of  the  point  P. 

The  co-ordinates  of  pojipts  are  designated  by  the  letters 
corresponding  to  the  co-ordinate  axes ;  that  is,  the  abscissas 
are  designated  by  the  letter  x,  and  the  ordinates  by  the 
letter  y. 

1.  K  the  co-ordinates  of  a  point  are  known,  the  position 
of  the  point  may  be  found.  For,  let  us  suppose  that  we 
know  the  co-ordinates  of  any  point,  as  I*.  Then,  from  the 
origin  A,  lay  off,  on  the  axis  of  abscissas,  a  distance  AD, 
equal  to  the  known  abscissa ;  and  through  D,  draw  a  par- 
allel to  the  axis  .of  ordinates.  Lay  off,  on  the  axis  of  ordi- 
nates, a  distance  AC,  equal  to  the  known  ordinate,  and 
through  C,  draw  a  parallel  to  the  axis  of  abscissas ;  the 
point  P,  in  which  it  meets  DD,  will  be  the  position  of  the 
point. 

2.  When  the  co-ordinates  of  a  point  are  kno^vn,  the  point 
is  said  to  be  give?i ;  and  we  have, 

X  =  a,    and    y  —  h ; 

these  are  called,  the  equations  of  the  point.    Hence, 

The  equations  of  a  poi7it  are  the  equations  which  ex- 
press the  distances  of  the  point  from  the  co-ordinate  axes. 


28  ANALYIICAL      GEOMETRY.  [bOOK   I. 

Second  Angle. 

6.    Let  US  consider  the  given  point  P',   in  the  second 
angle,   TAX'. 

Tlie    abscissa    of   this    point    is  y 

CP\   6r  AI)\  and  the   ordinate,  pi         I 

P'D\  or  AC.     Since  distances  es-  /        Jc 

timated  at  the  right  of   Y,   have      „,        /         j 

been   regarded   as  positive,  those  ^      M 

at  the  left,  are  nes^ative;*  hence,  / 

the  equations  of  the  point  P',  are, 

JB  =   —  a,    and    y  =  J. 


Third  Angle. 

7.  Let  US   consider  the    given  point    P",   in  the  third 
angle. 

The  abscissa  of  this  point  is 
CP",  or  AD\  and  negative. 
The  ordinate  is  D'P",  or  AC, 
Since  distances  above  the  axis  of  ~ 

-Zi    have  been  regarded  as  posi-  _ 

tive,  those  below  it  are  negative ;  ^"\ 

hence,  the  equations  of  the  point 
P',  are, 

JC  =  —  a,    and    y  =:  —  b. 

Fourth  Angle. 

8.  Let  us  consider  the   given  point  P"',  in  the  fourth 

angle. 

•  Bourdon,  Art.  89.     Universitv,  Art.  96. 


i 


BOOK  I.]  POINT      AND      STRAIGHT      LINE. 


29 


The  abscissa  of  this  point  is 
C'P'",  or  AD,  and  positive^ 
The  ordinate  is  DJP"',  or  A0\ 
and  negative ;  hence,  the  equa- 
tions of  the  point  jP"',  are, 


X  =  a,    and    y  = 


b. 


Therefore,  the  following  are  the  equations  of  a  point  in 
each  of  the  four  angles  : 


1st  angle, 
2d  angle, 
3d  angle, 
4th  angle. 


X  =  +a, 

X  —  —  a, 

X  =  —  a, 

X  =  +«, 


y  =  ■\-b. 
,V  =  +b. 

y  —  -b, 
y  —  —b. 


We  see,  by  examining  these  results,  that  the  signs  of  the 
abscissas  in  the  different  angles,  correspond  to  the  algebraic 
signs  of  the  cosines,  in  the  different  quadrants  of  the  circle  ; 
and  that  the  signs  of  the  ordinates,  correspond  to  the  alge- 
braic signs  of  the  sines.* 

^  EXAMPLES    IN    CONSTRUCTION. 

1.  Determine  the  point  whose  equations  are, 
jc  =r  3,     and    y  =  2. 


Having  drawn  the  co-ordinate 
axes,  lay  off,  on  the  axis  of  -X^  a 
distance  AB,  to  denote  the  unit 
of  length.  Then  lay  off,  from  A 
to  Z>,  three  times  the  unit  of 
length.     From  A,  on  the  axis  of 


X 
M 


*  Legendre,  Trigonometry,  Art.  69. 


30  *       ANALYTICAL      GKOMETEY.  [bOOK  I, 

y^  lay  off  a  distance  AJE^  equal  to  twice  the  unit  of  length. 
Through  D  and  JE^  draw  parallels  to  the  axes,  and  their 
point  of  intersection,  P,  will  be  the  required  point. 

2.  Determine  the  point  whose  co-ordinates  are, 

JB  =   —  5,     and    y  =  4. 

3.  Determine  the  point  whose  co-ordinates  are, 

a  =   —  V,     and    y  =   —  8. 

4.  Determine  the  point  whose  co-ordinates  are, 

ic  =  4,  and    y  =   —  6. 

Co-ordinate  Axes,  and  Origin. 

9.  Since  the  ordinate  of  a  point  is  its  distance  from  the 
axis  of  -Z",  the  ordinate  of  any  point  of  that  axis  must  be 
zero.  Hence,  the  equations  of  a  given  pointy  in  the  axis  of 
JXy  wiU  be, 

X  =   ±  a,    and    y  =  0 ; 

the"  plus  sign  before  a,  being  used  when  the  point  is  at  the 
right  of  the  origin,  and  the  minus  sign,  when  it  is  at  the 
left. 

If  we  attribute  to  «,  all  possible  values  between  0  and 
-f  00,  the  equations  will  embrace  all  points  of  the  axis  of 
-Zj  at  the  right  of  the  origin;  and  if  we  give  to  a,  all 
values  between  0  and  —  oo,  they  wiU  embrace  aU  points  of 
the  axis  of  -Z",  at  the  left  of  the  origin.  Both  these  coiv. 
ditions  are  expressed  by  the  simple  phrase, 
X  indeterminate. 

Hence,  for  all  points  in  the  axis  of  -ZJ  we  have, 
X  indeterminate,    and    y  =  0 


300K  I.]         POINT      AND      STBAIGHTLINE.  31 

10.  For  a  given  point  of  the  axis  of  Y^  we  have, 

a;  =  0,    and    y  =  ±h', 

the  plus  sign  before  J,  being  used  when  the  point  is  above 
the  axis  of  JT,  and  the  minus  sign  when  it  is  below.  For 
all  points  in  the  axis  of  Y^  we  have, 

a  =  0,    and    y  indeterminate. 

11.  Since  the   origin  of  co-ordinates  is  in  the  axis  of 

Yt  its  abscissa  is  zero;  and  since  it  is  in  the  axis  of  JT, 

its   ordinate    is    zero.      Hence,  the  equations  of  the  origin 

are, 

ic  =  0,     and    y  =  0. 

Straight  lines  in  the  same  plane. 

12.  T/ie  equation  of  a  line,  is  an  equation  which 
eocpresses  the  relation  between  the  co-ordinates  of  every  poi?U 
of  the  line,    - 

Equation  of  a  straight  line. 

13.  Let  A  be  the  origin  of 
co-ordinates,  and  AJC,  AY,  the 
co-ordinate  axes.  Through  A, 
draw  any  straight  line,  as  AP, 
making  with  the  axis  of  ^  an 
angle  denoted  by  a.  Denote  the 
angle  YAJT,  included  by  the  co- 
ordinate axes,  by  /3. 

Take  any  point  of  the  line,  as  P,  and  draw  PD  parallel 
to  the  axis  of  Y:  then,  PD  will  be  the  ordinate,  and 
AD  the   abscissa,  of  the  point  P, 


32 


ANALYTICAL      GEOMETEY. 


[nOOK   L 


Since  PB  is  parallel  to  the  axis  of  ordinates,  the  angle 
APD  is  equal  to   PAY\   that 
is,   equal  to    /3  —  a.  /  P 

Since  the  sides  of  a  triangle 
are  to  each  other  as  the  sines 
of  their  opposite  angles,*  we 
have 

PD  :  AB  : :  sm  a  :  sm  (/3  —  a). 

But  PB  is  to  AB^  as  y,  the  ordinate  ot  any  point  ot  the 
line  AP^  is  to  the  corresponding  abscissa  x\  therefore 


sin  a   :   sin  (/3  —  a), 


which  gives, 


y  = 


sm  a 


sin  {,3  —  a) 


«; 


and  this  is  the  equation  of  the  straight  line  AP,  sinco 
it  expresses  the  relation  between  the  co-ordinates  of  every 
point   of  the   line. 

1.  If  we  draw  a  Ime  parallel 
to  AP,  cutting  the  axis  of  Y 
at  a  distance  from  the  origin  de- 
noted by  5,  and  produce  the  or- 
dinate of  the  pomt  P  to  P\  it 
is  plain  that  the  ordinate  of  the 
point  P'  will  exceed  the  ordi- 
nate of  the  point  P  by  the  constant  quantity  b ;  hence,  the 
equation  of  the  parallel  line  will  be, 

sin  a 


sin  (3  —  a) 


X  -{-  b. 


*  Legendre,  Trig.  Art.  43. 


BOOK    I.]  POIT^T      AND      STRAIGHT      LINE. 


33 


2.  If  the  parallel  cuts  the  ax:^  Y  below  the  origin  of 
co-ordiiiates,  the  value  of  y,  in 
the  second  line,  will  be  less  than 
the  value  of  y  in  the  line  AP^ 
by  the  constant  quantity  b ;  and 
in  that  case,  b  becomes  nega- 
tive, and  the  general  equation 
takes  the  form. 


y 


sm  a 


sin  (?  -■  ri) 


3.  Since  the  line  PD  is  par- 
allel to  the  axis  of  T^  the  angle 
APD  is  equal  to  the  angle 
PA  Y\  hence,  the  coefficient  of 
X  is  the  sine  of  the  angle  ichich 
the  line  makes  with  the  axis  of 
X^   divided  by  the   sine  of  the 

angle  which  it  makes  with  the  axis  of  Y. 

4.  Thus  far,  we   have   supposed   the   co-ordinate   axes  to 
make  an   oblique   angle  with   each   other.      It   is,   however, 
generally,  most    convenient    to    refer    points    and    lines    to* 
co-ordinate   axes  which   are   at  riorht   anirles. 

If  we  suppose   YAX  to  be  a  right  angle, 

«• 

/3  -  a  =   90°  -  a, 
and,  sin  (^  —  a)  =  cos  a.* 


*  Legendre,  Trig.  Art.  63. 


34 


ANALYTICAL      GEOMETRY. 


[book  1. 


The  equation  of  the  straight  line  then  takes  the  form. 


y 


Bin  a 


X  ±h\ 


cos  a 

or,  y  =  tang  ax  ±  b, 

the  tangent  of  a  being  calculated  to  the  radius  1. 

If  we  denote  the  tangent  of  a  by  a,  the  equation  becomes, 
y  =  ax  ±  b. 


r 


f 

PK 

A 

/KY\ 

[/ 

x^^  — 

!x 

pin 

^  Interpretation  of  the  equation. 

14.  The  line  AP^  passing 
throucrh  the  orisrin  of  co-ordi- 
nates,  and  whose  equation  is 
y  =z  ax,  has  been  drawn  in  the 
first  angle.  But  the  equation  is 
equally  applicable  to  a  line 
drawn  in  either  of  the  other 
angles,  if  proper  values  and  signs  be  attributed  to  the  tan- 
gent, a.  The  angle,  of  which  a  is  the  tangent,  is  always 
estimated  from  the  axis  AX^^  around  to  the  left. 

1.  If  the  line  be  dra^\^l  in  the  first  angle,  the  tangent 
a  is  positive,*  and  the  co-ordinates  x  and  y,  are  both 
positive. 

2.  If  the  line  be  drawn  in  the  second  angle,  the  angle 
XAP  will  fall  in  the  second  quadrant,  and  its  tangent,  a, 
will  be  negative.*  But  the  abscissas  of  points  in  the  second 
angle  are  also  negative :  hence,  a  and  x  are  both  nega- 
tive: their  product  is,  therefore,  positive;   lieiice,  y  is  i  osi- 

*  Legendre,  Trig.  Art.  59. 


BOOK  I.]  POINT      AND      STRAIGHT      LINE.  35 

tive,  as  it  should  be,  since  it  represents  the  ordinates  of 
points  above  the  axis  of  abscissas. 

3.  If  the  line  AF"  be  drawn  in  the  third  angle,  the 
tangent  a  will  be  positive,  since  the  angle  falls  in  the 
third  quadrant  ;*  and  since  x  is  negative,  the  second  member 
will  be  negative ;  hence,  y  will  be  negative,  as  it  should  be. 

4.  If  the  line  AF'"  be  drawn  in  the  fourth  anffle,  the 
tangent  a  will  be  negative,  since  the  angle  falls  in  the  fourth 
quadrant ;  and  since  x  is  positive,  the  second  member  will 
be  negative,  and  therefore,  y  will  be  negative,  as  it 
should  be. 

As  the  same  reasoning  is  applicable  to  the  general  form, 

the   equation, 

y  =  aa;  +  J, 

will  represent  every  straight  line  w^hich  can  be  drawn  on 
the  plane  of  the  co-ordinate  axes,  if  proper  values  and  signs 
are  attributed  to  a  and  h, 

5.  The  values  of  a  and  h  are  constant  for  the  same 
straight  line,  but  take  different  values  when  we  pass  from 
one  line  to  another.  They  are  called  arbitrary  constants^ 
because  values  may  be  attributed  to  them  at  pleasure,  when 
only  the  form  of  the  equation  is  given. 

6.  If,  in  the  equation  of  a  straight  line 

y  =  ax  +  b, 

any  value  be  attributed  to  one  of  the  variables,  the  other 
becomes  determinate,  and  its  value  may  be  found  from  the 
equation. 

*  Legendre,  Trig.  Art.  69. 


36  AJTALYTICAL      GEOMETKT.  [bOOK  I. 

IfJ  for  example,  we  make, 

jc  =  1,  we  have,  y  =     a  +  b. 

SB  =  2,  gives,  y  =  2a  -^  b, 

a;  =  3,  gives,  y  =  3a  -{-  b, 

&c.,  &c.,                           &c. 

Or,   we    may  attribute  values  to  y,    and  find  the   corres- 
ponding values  of  x.    If  we  make. 


y  =  1. 

we  liare, 

y  =  2, 

gives, 

y  =  3, 

gives. 

1 

__ 

5 

X 

- 

a 

2 



i 

X 

— 

a 

<T» 

3 

— 

b 

Construction  of  Straight  Lines. 

15.  The  co7istruction  of  a  Hne,  represented  by  an  equa- 
tion, is  the  operation  of  drawing  the  line  on  the  plane  of 
the  co-ordinate  axes. 

16,  A  line  is  said  to  be  given,  or  Jcnown,  when  the 
form  of  its  equation  is  given,  and  when  the  constants  have 
Jixed  values.  The  position  of  the  line  is  then  determined, 
and  the  line  can  be  constructed. 

First  Method* 

1.  Construct  the  line  whose  equation  is, 

y  =  3ic. 

This  line  passes  through  the  origin  of  co-ordinates,  and 
makes  with  the  axis  of  ^  an  angle  whose  tangent  is  3. 
(Art.  13.) 


BOOK  I.] 


PROBLEMS. 


87 


-^ 


if   E 


Having  drawn  the  co-ordinate 
axes  at  right  angles  to  each 
other,  lay  off  from  the  origin, 
the  unit  of  length,  AB  =  1. 
At  B,  draw  BC  perpendicular 
to  the  axis  of  JT,  and  make  it 
equal  to  3  :  that  is,  to  three 
times  the  unit  of  length  ;  then,  draw  A  CF, 

Since  the  tangent  of  the  angle  at  the  base  is  equal  to 
the  perpendicular  divided  by  the  base,*  BO  will  be  the 
tangent  of  the  angle  BA  C,  to  the  radius  1 ;  hence,  A  CF 
is  the  required  line ;  for,  any  ordinate,  as  FE^  will  be 
equal  to  3  times  the  abscissa,  AE, 


2.  Construct  the  line  whose  equation  is, 
2/  =  2a;  +  4. 

This  line  will  cut  the  axis  of 
Y;  at  a  distance  from  the  ori- 
gin, above  the  axis  of  -Z^  equal 
to  4  (Art.  13).  Having  assumed 
the  unit  of  length,  lay  it  off  4 
times,  from  A  to  G.  Through 
(7,  draw  a  parallel  to  the  axis 
of  X,    Then  lay  off  CD,  equal 

to  the  unit  of  length.  Draw  BE  perpendicular  to  JT, 
and  make  it  equal  to  2.  The  indefinite  straight  line  pass- 
ing through  C  and  JEJ  will  be  the  line  required.  For,  since 
DE  is  twice  CZ>,  IIB  will  be  twice  CB  or  AF\  hence, 
any  ordinate,  as  EF^  will  be  equal  to  twice  the  abscissa, 
*  Legendre,  Trig.  Art.  37. 


3S 


ANALYTICAL      GEOMETRY. 


[book  I, 


AJFy  plus  AC,   which  is   4.      If  the   coefficient   of  x  were 
negative,   the   equation   would   take   the  form, 

y  =   —  2x  -\-  4; 

the  point  D  would  then  fall  at 
D',  and  the  line  would  take  the 
direction  C^\  Every  straight 
line  passing  through  the  origin 
of  co-ordinates  will  lie  in  the 
first  and  third  angles  when  the  co- 
efficient of  X  is  positive,  and  in  the 
second  and  fourth,  when  it  is  negative. 

If  4  were  negative,  the  point   G,  would  fall  on  the  axis 
of  T",  below  the  origin. 


\  "^ 

A 

!  /w 

^ 

X 

,  Second  Method. 

17.  A  point  which  is  common  to  two  straight  lines  wiU 
be  at  their  intersection ;  and  the  co-ordinates  of  this  point 
will   satisfy  the   equations   of  both  lines. 

Conversely,  if  the  equations  of  two  straight  lines  be  made 
simultaneous,*  and  combined,  the  results  obtained  will  be 
the  co-ordinates  of  the  common  point. 

1.   Construct  the  line  whose  equation  is, 

2/  =  -  6a;  -f  12    .     .     .  .     (1.) 

If  this  equation  be  combined  ^vith  the  equation  of  the 
axis  of  -Z",  which  is, 

X  indeterminate,     and  y  =  0     (Art.  9), 

we  shall  have,  a;  =  2, 

♦Bourdon,  Art.  82.      University,  Art  88. 


BOOK   I.]  PROBLEMS.  39 

which  is  the  abscissa  of  the  point  iu  which  the  line  inter- 
sects the  axis  of  JT. 

If  we   combine   Equation  ( 1 ),  with  the   equation   of   the 
axis  of  Y,  which  Ls, 

a;  =  0,     and    y  indeterminate  (Art.  lo), 

■we  shall  have,  y  =  12, 

which  is  the  ordinate  of  the  point  in  w^hich  the  line  inter- 
sects the  axis  of  Y, 

Having  drawn  the  co-ordinate  axes,  \J 

at  right  angles  to  each  other,  lay 
off,  on  the  axis  of  JT,  AJB  equal  to 
twice  the  unit  of  length  ;  and  on  Y, 
AC  equal  to  12  times  the  unit  of 
length,  and  then  draw  the  line  BC\  ~~^\  ^V" 
this  will  be  the   line   required,  since 

two  points  determine  the  position  of  a  straight  line.  The 
line  makes  an  obtuse  angle  with  the  axis  X,  as  it  should 
do,   since   the   coefficient  of  x  is  negative. 

2.   Construct  the  line  whose  equation  is, 

y  —  ax  +  h. 
S.   Construct  the  line  whose  equation  is, 

2/  =   2a;  -f  5. 
4.    Construct  the  line  whose  equation  is, 

y  =   —  X  —  I, 
0.   Construct  the  line  whose  equation  is, 

y  =:    _  2ic  -I-  6. 


4:0  ANALYTICAL      GEOMETRY.  [bOOK   I. 

Equation  of  the  first  degree  between  two  variables. 

18,     The  equation, 

Ay  +  Bx-{-  C  =  0, 

is  the  most  general  form  of  an  equation  of  the  first  degree 
between  two  variables,  since  there  is  an  absolute  term    (7, 
and  since  each  of  the  variables,  y  and  ic,  has  a  coefficient. 
This  equation  may  be  written  under  the  form, 

B         C 

which  becomes  of  the  form  already  discussed,  if  we  make, 

7  =  «,     and 7  =  J. 

A  A 

Havinj?    dra\\Ti   the   co-ordinate   axes   at    ripjht   ano:lea  to 

each  other,  if  we  lay  off  on  the  axis  of  !Fi  a  distance  equal 

(J 
to    —  -j-i   ^Q<i    through  the    point  so    determined,  draw   a 

line  which  shall  make  with  the  axis  of  JT  an  angle  whose 
tangent  is  —  -j  ;  it  mtlII  be  the  straight  line  whose  equa- 
tion is, 

Ay  -{-  Bx  +  C  =  0. 

We  may  also  put  the  equation  under  the  form, 

A         C 

^=  "Wb^ 

A 

in   which ^    is  the   tangent    of  the    angle    which    the 

C 
straight  line  makes  with  the  axis  of  J^  and  —  —  the  dis- 
tance cut    off  on   the   axis  of  JT,  measured  from  the  origin. 


BOOK    I.]         POINT      AND      STRAIGHT      LINE. 


41 


Hence,  the  following  principles : 

1.  Every  equation  of  the  first  degree^  between  two  variables^ 
is  the  equation  of  a  straight  Ihie : 

2.  ^  the  equation  be  solved  with  reference  to  y,  the  co- 
efficient of  X  will  denote  the  tangent  of  the  angle  which  th^ 
line  makes  with  the  axis  of  JC;  and  the  absolute  term 
will  denote  the  distance  cut  off  on  the  axis  of  Y:  and  sim- 
ilarly^ if  the  equation  be  solved  with  reference  to  x. 


Distance  between  two  points. 

19.  A  point  is  said  to  be  given,  when  its  co-ordinates 
are  known.  I^own  co-ordinates  are  usually  designated  by 
marking  the  letters,  thus  : 

X',  y'  ;      X",  y"  ;      x'",  y'"  ; 

which  are  read,  x  prime,  y  prime,  x  second,  y  second,  &g. 

Let  If  and  JV  be  two   given  y 

points.  Designate  the  co-ordinates 
of  J/,  by  x\  y\  and  the  co-ordi- 
nates of  K^  by  x'\  y'\  and  il/iV, 
the  distance  between  them,  by  D. 
Then, 


But, 

hence, 


MP  =  x"  —  x\    and    JSfP  =  y"  ■ 

Mlf  =  MP"  +  Plf ; 
i)2  =       {x"  -  x'Y  -I-  {y"  -  y'Y 


or, 


D    =   y^{x"  -  x'Y  +  (y"  -  y'Y;  that  is, 

The  distance  between  any  two  points  is  equal  to  tJie 
square  root  of  the  sum  of  the  squares  of  the  difference 
of  their  abscissas  and  ordinates. 


4i5  ANALYTICAL      GEOilETEY.  [bOOK   L 

1.    If  either  of  the  points,  as  M^   coincides  with  the  ori- 
gin,  its  equations  will  become, 

x'  =  0,    and    y'  =  0, 

xnd  we  shall  have, 


J)  =   -^x"^  +  y"\ 
a  result  which  may  be  easily  veiified. 

*        Equation  of  a  line  passing  through  a  given  point. 

20.     Let  ^  be   a  given  point,  y  \My 

and  designate  its  co-ordinates  by 
m'y  y'.  It  is  required  to  find  the 
equation  of  a  line  which  shall  pass 
through  this  point. 

The  equation  of  every  straight  line  is  of  the  form, 

y  =  ax  ^h (I.) 

Since  the  required  line  is  to  pass  through  the  point  M^ 
tlie  co-ordinates  of  that  point  must  satisfy  Equation  ( 1 ) ; 
hence,  we  have, 

y'  =  ax'  ■\-l (2.) 

Combining  Equations  ( 1 )  and  { 2 ),  we  obtain, 

y  —  y'  =  a{x  —  x'), 

which  is  the  equation  of  any  line  passing  through  the  point 
whose  co-ordinates  are  x'  and  y', 

1.  In  this  equation,  the  tangent,  cr,  is  undetermined,  as 
it  should  be,  since  an  infinite  number  of  lines  may  be  drawn 
through  the  point  M, 


BOOK  I.J         POINT      AND      STEAIGHT      LINE.  43 

Equation  of  a  line  passing  through  two  given  points. 

81,     Let  M  and  N'^  be  two  giv-  y  jf 

en  points.  Designate  the  co-ordi- 
nates of  the  first  by  x\  y\  and 
the  co-ordinates  of  the  second  by 

Since    the    required    line    is    to 

pass  through  the  point  Jf,  whose 

co-ordhiates  are    x\  y\  its  equation  will    be  of  the  form 

(Art.  20), 

y  -  y'  =  «(aj  -  ax')     ....    (1.) 

in  which  x  and  y  are  the  co-ordinates  of  every  point  of 
the  line. 

Since  the  required  line  is  also  to  pass  through  the  pomt 
Ny  whose  co-ordinates  are  x'\  y'\  these  co-ordinates,  when 
substituted  for  x  and  y,  in  Equation  (1),  will  satisfy  that 
equation  ;  hence,  we  have, 

y"  -  y'  =z  a{x"  —  x')-, 

from  which  we  find,  in  known  quantities, 

y"  -  y' 


a  = 


x"  -  x'        •    ■    •    • 

Substituting  this  value  in  Equation  ( 1 ),  we  have, 


?/"  -  ?/■ 


J'-2''  =  S^^(^-'^')' 


(2.) 


(3.) 


which  is  the  equation  of  the  required  line. 

Had  we  first  passed  the  line  through  the  point  whose 
co-ordinates  are  x'\  y",  Equation  (1)  would  have  taken 
the  form, 

y  -~  y"  =  a{x  —  x") ; 


44  ANALYTICAL      GE0:METRT.  [bOOK  1. 

and  tlie  equation  of  the  required  line  would  have  been, 

y-y"  =  i^'^''-^"'>  •  •  •  (*•> 

1.  The  value  of  a,  found  in  Equa-         y  jf 
tion    ( 2 ),    is    easily    verified.      For, 
y"  —  y'  is  equal  to  iVP,  and  x"  —  x' 
is  equal  to  J/P;  hence, 

^  _  y"  -  y' 
MP  ~  x"  -  x' ' 

and,  consequently,  equal  to  the  tangent  of  the  angle  NMP^ 
to  the  radius  1.*  Hence,  a  line  passing  through  either  of 
the  points  M  or  iV,  Equations  (3)  and  (4),  and  making 
with  the  axis  of  JT,  an  angle  whose  tangent  is  NP  -v-  MP^ 
will  also  pass  through  the  other  point. 

2.  If,  in  Equation  (2),  we  suppose         y 


y' 


y'\    we  shall  have, 
0 


a  = 


X' 


X 


>  =  0; 


M 


JV 


and    this  is  as    it  should    be,   since 

under  this  supposition,  the  line  becomes  parallel  to  the  axis 

of  JT. 

3.  If  we  suppose  a'  =  jb",  in 
Equation  (2),  the  ordinates  y'  and 
y"  being  unequal,  we  shall  have. 


y 


y' 


A 


N 

J 

i 

AT 


tlierefore,   a  is  infinite;!   and  hence, 

the  line  is  perpendicular  to  the  axis  of -Z!  J 

•Leg.,  Tr.  Art.  37.        f  ^y  ^rt.  71.     Un.,  Art.  72.        X  Leg.,  Tr.  Art  60. 


BOOK   I.]         POINT      AND      STRAIGHT      LINE.  45 

If  we  suppose  y'  =  y'\  and  at  the  same  time,  x'  =  ar"', 
the  two  points  will  coincide,  and  we  shall  have, 

0 

Under  these  suppositions,  a  is  indeterminate,*  as  it  should 
be,  since  an  infinite  number  of  straight  lines  may  be  drawn 
through  a  single  point. 


Equation  of  a  line  parallel  to  a  given  line. 

22.    Let  y  =  ax  -^  h 

be  the  equation   of  a  given  line  (Art!  16). 
The   equation   of  the  required  line  will  be  of  the   form 

in  which  a*  and  5',   are  undetermined. 

Two  right  lines  are  parallel,  when  they  make  the  same 
angle  with  the  axis  of  abscissas.    Hence,   if  we  make, 

a'  —  a, 

the  second  line  will  be  parallel  to  the  first ;  and  its  equa- 
tion will  be, 

y  =z  ax  -\-  h\ 

in  which  equation,  V  is  undetermined,  as  it  should  be, 
since  an  infinite  number  of  lines  may  be  drawni  parallel 
to   a  given  line. 

1.    If  it  be  required  that  the  parallel  shall  pass  through 
a  given  point,  its  position  will  be   entirely  determined. 
For,  if  the   co-ordinates  of  the  given  point  be   denoted 
•Bourdon,  Art.  71.     University,  Art.  72. 


46 


ANALYTICAL      GEOMETRY. 


[book  I. 


b7  a'   and  y',  the   equation  of  the  parallel  will  take  the 
form   (Art.  21), 

1/  -y'  =  a{x-  x'), 

in  which  the  quantities  a,  y\  x\  are   known;  hence,  the 
position   of  the  line  is  fixed. 

Angle  included  between  two  given  lines. 
« 
23.    Let  DC,  BC.he  the 

two   given  lines: 

y  =  ax  +  b, 
tiie  equation  of  the  1st, 

y  =  a'x  +  b\ 

the  equation  of  the  2d, 

in  which   cr,    a',    b,    b\  are  known. 

Denote  the   angles   CDX  and  CBX,  by  a  and  a',  and 
the   angle  DCB,  by  V, 

Then,  since       CBX  =   CDB  +  DCB* 

we  have,  F"  =  a'  —  a, 

and,      tanff  y  =  tangr  (a.'  —  a)  =  2. & — 

^  "=  ^  ''         1  +  tang  a'  tang  a 

to   the   radius   l.f 

Substituting    for    tang  a',    and    tang  a,  their    values  a' 
and  a,   we  have, 

TT        a'  —  a 

^  1  +  a'a  ••; 


•  Legendre,  Bk.  I.    Prop.  25.   Cor.  6.  f  Trig.  Art.  66. 


BOOK  I.]  POINT      AND      STRAIGHT      LINE, 


47 


1.    If  the  lines  becomo  parallel,  the  angle   V  will  be  0, 


and  hence. 


tanor  y  — 


a'  —  a 


=  0. 


1  +  aa 

Therefore,  a'  —  a  =  0;    or,    a'  =  «, 

a  relation  already  proved  (Art.  22). 

2.    If  the  lines  are  perpendicular  to  each  other,   "Fifill 
be   equal  to   90°,   and  its  tangent  infinite,!   that  is, 

a'  —  a 


hence, 


tang  V  =  r— — -  =  00  ;  t 
1  +  a'a  =  0. 


This  last,  is  the  equation  of  condition,  when  two  ri^t 
lilies  are  at  right  angles  to  each  other.  If  the  tangent 
of  one  of  the  angles  is  known,  the  other  can  be  found 
fi'om  the   equation  of  condition. 

Intersection  of  two  lines. 
24.     Let      y  =  ax  -\-  b,      be  the  equation  of  the  first; 
and  y  =  a'x  +  b\    the  equation  of  the   second. 

The  pomt  in  which  two 
straight  lines  intersect  each 
other,  is  found,  at  the  same 
time,  on  both  of  the  lines; 
hence,  its  co-ordinates  will 
satisfy  both  equations. 

If,  therefore,  we  suppose  y 

•  Trig.  Art.  60.     f  Leg.,  Trig.  Art.  60     X  Bour.,  Art.  71.  Univ.  Art.  72. 


ot-^   / 


Ay-. 


-.  «v 


'U' 


43 


ANALYTICAL       GEOMETliY.  [bOOK   L 


and  a',  in  the  equation  of  the 
first  line  to  become  equal  to 
y  and  a*,  in  the  equation  of 
the  second,  the  two  equations 
will  be  simultaneous. 

Combining  the  equations, 
mider  this  suj^position,  and 
designating  the  co-ordinates 
Df  the  point  of  intersection,  by  a'  and  y\  we  find. 


a  —  a' 


and    y'  = 


ah'  -  a'h 


a 


a' 


1.  If,  in  the  two  last  equations,  we  suppose  a  =  a', 
the  values  of  x'  and  y'  will  both  become  infinite.  The 
supposition  of  a  =  a',  is  the  condition  when  the  two  lines 
are  parallel ;  and  therefore,  under  this  supposition,  their  point 
of  intersection  ought  to  be  at  an  infinite  distance  fi-om 
both   the   co-ordinate   axes. 

If,  at  the  same  time,  we  suppose  5  =  5',  the  values 
of  x'  and  y',  will  become  equal  to  0  di\dded  by  0,  that 
is,  indeterminate.  The  two  suppositions  will  cause  the  lines 
to  coincide;  hence,  their  point  of  intersection  ought  to  be 
indeterminate,  since  eveiy  point  of  either  line  will  satisfy 
both   equations. 


A  perpendicular  from  a  given  point  to  a  given  line. 

25.    Let  y  =  a^  +  5, (1.) 

be   the    equation  of  a  given  line,   and   cc',   y',   the   co-ordi. 
nates  of  a  given  point. 


BOOK   I.]        POINT      AND      STRAIGHT      LINE.  49 

It  is  required  to  draw  from  this  point,  a  line  perpen- 
dicular to  the  given  line,  and  to  find  the  length  of  the 
perpendicular. 

The  equation  of  a  line  passing  through  a  point,  whose 
co-ordinates  are  x\  y\  (Art.  20),  is 

y  -y'  =  «'(^  -  aj'),  ....    (2.) 

in   which   a'  denotes  the   tangent   of  the    angle  which   this 
second  line  makes  vat\\  the  axis  of  A". 

But  since  this  line  is  to  be  perpendicular  to  the  given 
line,   we   have    (Art.  23),  y  * 


1  + 

aa'  = 

=  0; 

from  which 

we 

have. 

• 

o' 

=z      - 

1 
a 

Substituting  this  value  for  a',  in  Equation  (2),  the  sec- 
ond line  becomes   perpendicular  to  the  first,  and   we  have,. 

y  -  y'  =   -  -i^  -  ^')     '     .     .      (3.) 

It  is  now  required  to  find  the  length  of  the  perpen-- 
dicular. 

This  is  done,  1st,  by  finding  the  co-ordinates  of  the  point 
in  which  the  perpendicular  intersects  the  given  line;  and 
2d,  by  finding  the  differences  between  the  co-ordinates  of" 
this  point  and  the  co-ordinates  of  the  given  point ;  3d,  by 
substituting  these  differences  in  Formula  (Art.  19). 

Let   us   designate  the   co-ordinates  of  the  point  of  inter- 
section, by  ic",  y'\    Then,  since  the  pohit   is   on   the  given 
4 


50  ANALYTICAL      GEOMBlTET.  [bOOK  U 

line,  its  co-ordinates  will  satisfy  the  equation  of  that  line, 
and   we   shall   have, 

y"  =  ax"  +h^ (4.) 

and  since  the  point  is  also  on  the  perpendicular,  its  co. 
ordmates  ^ill  also  satisfy  the  equation  of  the  pei-pendicular 
and  give, 

y"-y'  =  -Ux"-x')     .    .    .    (5.) 

(J/ 

If  we   eliminate  x'\  from  these  two   equations,  we  shall 

have, 

,,  _  a^y'  4-  Gx'  +  h 
^    ~         1  +  a2 

Subtracting  y'  from  both  members,   we  obtain, 

y'  —  ax'  —  h  ,    . 

Substituting   this  value    of    y"  —  y\    in    Equation   { 5 ), 

we  have, 

a{y'  —  ax'—  h) 

1  +  a" 


X"  -X'  ^   ^     ^\    ,      . ^.     .     .     .     (7.) 


Let  us  designate  the  length  of  the  perpendicular,  by  P. 
Since  the  distance  between  two  points,  whose  co-ordinates 
are,    x",  y'\    x\  y\     (Art.  19),  is 


we  have,  by  substituting  for    x"  —  x',    and   y"  —  y',   their 
values  found  in  Equations   (7)   and   (6), 

jp  —  y'  -  ^^'  -f> . 

*1.    If  the  given    point    should    fall    on   the   given    line, 


BOOK  I.]      TRANSFORMATION     OF     CO-ORDINATES 


51 


its  co-ordinates  would  satisfy  the  equation  of  the  line,  and 

give, 

y'  =  ax'  +  5. 

This  supposition  would  reduce  the  numerator  of  the  value 
of  jP,  to  0,  and  consequently,  P  would  be  equal  to  0. 


A 


ly 


TRANSFOEMATION      OF      CO-OEDrNATES. 

26.  The  equations  of  a  point  determine  its  position 
with  respect  to  the  co-ordinate  axes  (Art.  5).  The  co- 
ordinate axes  may  be  selected  at  pleasure,  and  any  point 
may,  at  the  same  time,  be  referred  to  several  sets  of 
axes. 

Let  A^  for  example,  be  the 
origia  of  a  system  of  co-ordinate 
axes,  and  A\  any  point  whose 
co-ordinates  are  a  and  b. 

Through  A\  draw  two  new 
axes,  respectively  parallel  to  the 
first. 

The  co-ordinates  of  any  point,  as  P,  referred  to  the 
primitive  system,  are  AD^  PD ;  and  its  co-ordinates  referred 
to  the  new  system,  are  A'D\  PD'.  The  point  P  is  equally 
determined,  to  which  ever  system  it  be  referred. 

27.  It  is  often  necessary,  for  reasons  that  will  be  here- 
^er  explained,  to  change  the  reference  of  points  from  one 
system  of  co-ordinate  axes  to  another.  This  is  called,  tlie 
transformation  of  co-ordinates.  The  axes  to  which  the 
points  are  first  referred,  are  called.  Primitive  Axes  /  *and 
the  second  axes,  to  which  they  are  referred,  are  called, 
New  Axes, 


52  ANALYTICAL      GEOMETRY.  [bOOK  I. 

In  changing  the  reference  of  points  from  one  system  to 
another,  all  that  is  necessary,  is  to  find  the  co-ordinates  of 
the  points  referred  to  the  primitive  axes,  in  terms  of  the 
co-ordinates  of  the  new  origin  and  the  co-ordinates  of  the 
points  referred  to  the  new  axes. 

To  pass  from  a  system  of  co-ordinate  axes,  to  a  parallel  system, 

28.     Let  A  be  the  origin  of  the  Y  y 

primitive  system,  and  A'  the  origin 
of  the  new  system.  Suppose  the 
co-ordinates  of  the  origin  A\  to 
be,  AB  =  a,  and  JBA'  =  b;  and 
let  us  designate  the  new  axes,  by 
JC'  and  Y\  and  the  co-ordinates 
of  any  point,  referred  to  these  axes,  by  x'  and  y'. 

Then,  assuming  any  point,  as  JP,  we  shall  have, 

AB  =  AB  -f  BB,    and    BB  =  BB'  -f  B'P, 

Now,  since  AB  is  the  abscissa  of  P,  and  BB  =  A'B\ 
its  abscissa  referred  to  the  new  axes ;  and  since  BB  is  the 
ordinate  of  P,  and  B'B  its  ordinate  referred  to  the  new 
axes,  we  have, 

X  —  a  '\-  x\     and    y  =  h  -\-  y\ 

in  which,  the  primitive  co-ordinates  of  any  point,  are  ex- 
pressed in  terms  of  the  co-ordinates  of  the  new  origin,  and 
the  new  co-ordinates  of  the  same  point. 

1.  The  new  origin  may  be  placed  in  either  of  the  four 
angles  of  the  primitive  axes,  by  attributing  proper  signs  to 
its  co-ordinates,   a  and  h.     It  is  also  to  be  observed,  that 


BOOK  I.]     TRANSFORMATION     OF    CO-ORDINATES 


x'  and  y'  have  the  same  algebraic  signs,  in  the  different 
angles  of  the  new  system,  as  have  been  attributed  to  x  and 
y,  in  the  corresponding  angles  of  the  primitive  system. 


^ 


To  pass  from  a  rectangular  to  an  oblique  system. 


29.  Let  A  be  the  common 
origin,  AX^  AY^  the  primitive 
axes,  and  AX!^  AY\  the  new 
axes ;  and  let  ns  designate,  as 
before,  the  co-ordinates  of  points 
referred  to  the  new  axes,  by  x' 
and  2/'. 

Denote  the  angle  which  the 
new  axis  of  X  makes  with  the  primitive  axis  of  X,  by  a, 
and  the  angle  which  Y  makes  with  AX,  by  a' ;  and  let  P 
be  any  point  in  the  plane  of  the  axes.  Through  P,  draw 
Pi?  parallel  to  the  axis  of  Y,  and  PF'  parallel  to  the  axis 
of  Y' ;  draw  also  P'P  parallel  to  the  axis  of  Y,  and 
P'(7  parallel  to   the   axis   of  X. 

Then,  AB  =  AR  -f  PP, 

"will  be  the  abscissa  of  P,  referred  to  the  prunitive  axes ; 

and,  FB  =  J3G  +  CP, 

will  be  its  ordinate. 

Also,  AP'  will  be  the  abscissa  of  P,  referred  to  the  new 
system;  then  PP^  will  be  its  ordinate. 

But,  AR  =  AP'  cos  a,* 

that  is,  AR  =  x'  cos  a. 


and, 


RB 


P'C  =  PP' cos  a' 


y'  cos  a' ; 


Trig.  Art.  87. 


54  ANALYTICAL      GEOMETBY.  [bOOK  I. 

hence,  a  =  a'  cos  a,  -{-  y'  cos  a'. 

We  also  have, 

jP'H  =   CJ3  =z  AP'  sin  a  ; 
that  is,  €B  =  a'  sin  a,    and, 

FC  =  Fr  sin  a'  =  y'  sin  a' ; 

hence,      y  =  a'  sin  a  -f  y'  sin  a'. 

Hence,  the  formulas  are, 
X  =  x^  cos  a  4-  2/'  COS  a',         y  =  a'  sin  a  -{-  2/'  sin  a'. 

1.  If  it  were  required,  at  the  same  time,  to  change  the 
origin,  to  a  point  whose  co-ordinates,  referred  to  the  prim- 
itive system,  are  a  and  5,  the  formulas  would  become, 

X  =  a  -{-  x'  cos  a  -f  y'  cos  a',      y  =  b  -{-  x'  sma  -\-  y'  sin  a'. 

2.  If  a'  —  a  =  90°,  we  have,  sin  a'  =  cos  a  ; 
cos  a'  =  —  sin  a ;  *  substituting  these  values  in  the  last 
equation,  we  have, 

X  =  a  •{•  x'  cos  a  —  2/'  sin  a,      y  =  b  -{•  x'  sma  -{-  y'  cos  a, 

which   are  the  formulas  for  passing  from  one  system  of 
rectangular  co-ordinate  axes  to  another. 


To  pass  from  an  oblique  to  a  rectangular  system. 

30.    The  first  set  of  formulas  of  the  last  Article,  for  a 
common  origin,  are, 

X  =  x^  cos  a  +  2/'  cos  aj         y  •=  x'  sin  a  -f  y'  sin  a'. 

*  Trig.  Art  63.  ' 


BOOK   I.J     TRANSFORMATION     OF    CO-ORDINATES.  55 

If  "we  regard  the  oblique  as  the  primitive  axes,  it  be- 
comes necessary  to  find  the  co-ordinates  of  points  referred 
to  these  axes,  in  terms  of  the  rectangular  co-ordinates,  and 
the  angles  a  and  a' ;  that  is,  we  must  find  the  values  of 
x'  and  y'. 

If  we  multiply  both  members  of  the  first  equation  by  the 
sin  cl\  and  both  members  of  the  second,  by  cos  a',  and  then 
subtract    them,     and     remember     that,     sin     (a'   —   a)   = 
sin  a!  cos  a  —  sin  a  cos  a',  y'  will  be  ehminated ;   and   if  x 
be  eliminated,    in  a  similar  manner,  we  shall  obtain, 

,  _  a;  sin  a'  —  y  cos  a'         t  _  V  ^^^  a  —  a;  sin  a 
sin  (a'  —  a)         >      ^  gjj^  ^a'  —  a) 

1.  K  the  origin  be  changed,  at  the  same  time,  to  a  point 
whose  co-ordinates,  with  reference  to  the  oblique  system, 
are  a  and   J,   we   shall  have, 

.   ic  sin  a'  —  y  cos  aJ         ,        -,        V  cos  a  —  a;  sin  a 
sm  (a'  —  a)        '     ^  sin  (a'  —  a) 

REMARKS. 

31.  The  primitive  co-ordinates  of  any  point,  determined 
with  reference  to  a  new  system,  depend  for  their  values, 

Ist.   On  the  position  of  the  new  origin: 

2d.  On  the  angles  which  the  new  axes  make  with  the 
primitive  axes  :   iind, 

3d.  On  the  co-ordinates  of  the  same  point,  referred  to 
the  new  system. 

32.  The    transformation    of   co-ordinates   embraces    two 

distinct    classes   of  propositions: 

1st.  To  transfer  the  reference  of  points  from  one  system 
of  co-ordinate  axes  to  another  system,  which  is  known.  In 
this    case,   the    co-ordinates    of   the    new    origin,   and    the 


I 


56  ANALYTICAL      GEOMETRY.  [boOK    I. 

angles  which  the  new  axes  make  with  the  primitive  axes, 
are   known. 

2d.  So  to  dispose  of  the  new  origin,  and  to  give  such 
directions  to  the  new  axes,  as  to  cause  the  resulting  equar 
tions  to  fulfil  certain  conditions,  or  to  assume  certain  forms. 
In  this  case,  the  conditions  imposed,  determine  the  position 
of  the  new  origin,  and  the  directions  of  the  new  axes. 

33.  Since  the  primitive  co-ordinates  of  points  are  always 
deteiTnined  in  linear  functions  of  the  new  co-ordinates,  that 
is,  by  equations  of  the  first  degree,  the  substitution  of  their 
A^alues  in  the  equation  of  any  line,  will  not  alter  the  degree 
of  that  equation ;  hence, 

A  given  equatioii  of  a  line^  and  its  equation  ichen  re- 
ferred to  a  new  system  of  co-ordinate  axes^  will  always  he 
of  the  sa)ne  degree. 

34.  We  shall  terminate  this  subject  by  a  single  example. 
Having  given,  the  equation  of  a  straight  line, 

y  z=z  a'x  +  h\ 

referred  to  rectangular  co-ordinates,  it  is  required  to  find 
its  equation  when  the  line  is  referred  to  oblique  co-ordi- 
nates having  a  difierent  origin.     We  have  (Art.  29 — 1), 

X  rr:  a  4-  a'  cos  a  -f  2/'  COS  a',       y  =  h  +  x'  mi  a  -\-  y'  sin  a'. 

Substituting  these  values  for  x  and  y,  in  the  equation  of 
the  line,  we  have, 
5  4-  x'  sin  a  4-  y'  sin  a'  =  a' {a  +  x'  cos  a  4-  y'  cos  a'J  4-  h' ; 

or,  by  reducing, 

,         a'  cos  a  —  sin  a     ,  aa'  -'-  h'  —  h 

^   ~  sin  a'  —  a'  cos  a'  sin  a'  —  a'  cos  a' ' 


BOOK    I.]  POLAR      CO-OEDIK^TES.  57 

wliich  is  the  equation  of  the  straight  line,  referred  to  the 
oblique  axes.  The  coefficient  of  a;',  is  the  sine  of  the  angle 
which  the  line  makes  with  the  axis  of  JT',  divided  by  the 
sine  of  the  angle  which  it  makes  with  the  axis  of  Y' 
(Art.  13).  The  second  term,  in  the  second  member,  is 
the  distance  cut  off  from  the  axis   of  Y'  (Art.  13 — 3). 

POLAR     CO-ORDINATES. 

85.  We  have  seen,  that  the  relative  position  of  points 
and  lines  may  be  determined,  analytically,  by  referring  them 
to  two  co-ordinate  axes.  There  are  also  other  methods,  by 
which  they  may  likewise  be  detei-mined. 

Assume,  for  example,  any  point,  as 
A,  and  through  it  draw  any  straight 
line,  as  AX^.    If  we  suppose  a  straight  y^ 

line,  as  AJS,  to  be  turned  around  the  ^^\ 

point  A,  so  as  to  make  with  AJC  all       ^ ■ 

possible   angles,  from   0   to   3G0°,  and 

suppose,  at  the  same  time,  the  line  AB  to  increase  or 
dimmish  at  pleasure,  the  extremity  Jj,  may  be  made  to 
occupy,  in  succession,  every  point  of  the  plane. 

Under  this  hypothesis,  there  are  two  variable  quantities 
considered:  1st,  the  variable  angle  XAJ3;  and  2d,  the  va- 
riable distance  A£'y  and  every  point,  in  the  plane,  may 
be  determined  by  attributing  suitable  values  to  these  vari- 
ables. 

The  fixed  line  AJC,  is  called  the  initial  line;  the  fixed 
point  A^  the  pole;  XAB  the  variable  angle^  and  AB  the 
radius-vector.  This  method  of  determining  the  position  of 
points,  is  called  the  systerii  of  polar  co-ordinates. 


I 


I 


58 


ANALYTICAL      GEOMETKY, 


Designate  the  variable  angle  JCAJ^, 
by  V,  the  radius-vector  AJS,  by  r, 
and  the  co-ordinates  of  the  point  ^, 
referred  to  rectangular  axes,  by  x 
and  y ;  then,  if  the  origin '  of  the 
rectangular  axes  be  at  A^  we  shall 
have, 


X  —  r  cos  V, 

and,  y  =  r  sin  v.* 

From  the  first  equation,  we  have, 


[book  I. 


r  = 


cos  V 


Now,  since  x,  and  the  cos  v,  are  both  positive  in  the  first 
and  fourth  angles,  and  both  negative  in  the  second  and 
third,  they  will  always  be  afiected  with  the  same  sign  ; 
and  hence,  the  sign  of  r  will  be  constantly  positive ;  con- 
sequently, 

A  negative  value  of  the  radius-vector  can  never  enter 
into  the  analysis. 

If,  therefore,  such  a  value  should  be  obtained,  we  infer, 
that  incompatible  conditions  have  been  introduced  into  the 
equations  ;  and  hence,  all  negative  values  of  the  radium 
vector  must  he  rejected. 


To  pass  from  a  rectangular  to  a  polar  system. 

36.    Let  A  be  the  origin  of  the  co-ordinate  axes.  A'  the 
pole,  A'B^   parallel  to  AX^  the  initial  line. 
Legendre,  Trig.  37. 


BOOK   I.J 


POLAR      CO-ORDINATES. 


59 


Then,  the  line  AF  will  be  the 
radius-vector  of  the  point  P.  Let 
the  co-ordinates  of  the  pole  A',  be 
denoted  by  a  and  b. 


Then, 
and, 
But, 
and, 
hence, 
and, 


D  X 


A'R  =  r  cos  V, 
Pi2  =  r  sin  v. 
AD  =  AB  +  BB, 

FB  =  BB  +  BB; 

X  =  a  -\-  r  cos  Vj 
y  z=  6  +  r  sin  V ; 
which  are  the  required  formulas. 

1.  If  the  pole  A\  be  placed  at  the  origin  A^  the  equa- 
tions will  become, 

X  =z  r  cos  V,        y  =:  r  sin  v. 

2.  If,  instead  of  estimating  the 
variable  angle  v  from  the  initial 
line  A'R^  parallel  to  AJC^  it  be 
estimated  from  A'R\  making  with 
AX  a  given  angle,  ±a,  the  equa- 
tions will  become, 

X  =  a  -\-  r  cos  {v  ±  a), 

y  =  b  +  r  sm  {v  dz  a). 


BOOK    II. 

OF      THE        CIKCLE. 

1.  The  equation  of  a  line  expresses  the  relation  which 
exists  between  the  co-ordinates  of  every 'point  of  the  line 
(Art.  12). 

2.  Lines  are  divided  into  different  orders,  according  to 
the  degree  of  their  equations.  For  example,  the  right  line 
is  a  line  of  the  first  order,  since  its  equation  is  of  the  first 
degree.  The  circumference  of  the  circle  is  a  line  of  the 
second  order,  its  equation  being  of  the  second  degree; 
and  if  the  equation  of  a  line  were  of  the  third  degree, 
the  line  would  be   of  the   thu-d  order. 

3.  Tlie  Interpretation  of  an  equation,  consists  in.  class- 
ing the  line  which  the  equation  represents;  in  determining 
its  position,  its  form,  its  limits,  and  the  points  iu  which 
it   intersects  the   co-ordinate   axes. 

Equation  of  the  circvunference  of  a  circle. 

4.  Let  A  be  the  origm  of  co- 
ordinates, and  ^l-3r,  A  Y,  the  co- 
ordinate  axes. 

It  is  required  to  find  the  equa- 
tion of  a  curve  such,  that  all  its 
points  shall  be  at  a  given  dist- 
ance from  the  origin  A.      Let  i? 


BOOK  n.J 


THE      CIRCLE. 


61 


denote  the  distance,  and   x  and  y,  the   co-ordinates  of  any 
point   of  the   curve,   as   P.     The 
square   of  the   distance  from  the 
origin    to     any  point,  whose    co- 
ordinates are  x  and  y,  is, 
a;2  ^  2/2 . 

hence,  x^  -\-  y^  —  i?^, 

which  is  the  equation  required. 


Interpretation. 

1,  To  interpret  the  equation,  we  begin  by  determining 
the  points  in  which  the  circumference  cuts  the  co-ordinate 
axes. 

The  co-ordinates  of  these  points  must  satisfy,  at  the 
game  time,  both  the  equation  of  the  circle,  and  the  equa- 
tions of  the   axes. 

The  equations  of  the  axis  of  X  (Bk.  L,  Art.  9),  being 

X  indeterminate,     and    y  =  0 ; 

if  we    make    y  =  0,     in  the   equation   of   the   circle,   the 

corresponding   values   of  x  will   be   the    abscissas    of  those 

points  which    are    common    to    the  circumference   and  the 

axis  of  JT;  that  is, 

X  ^   ±R\ 

which  shows  that  the  curve  cuts  the  axis  of  abscissas  in 
two  points,  one  on  each  side  of  the  origin,  and  each  at  a 
distance  from  it  equal  to  the  radius   of  the   circle. 

2.  To  find  the  points  in  which  the  circumference  cuts 
the  axis   3^  make    ic  =  0,    and  there  results, 

2/  =  ±i?; 


62  ANALYTICAL      GEOilETEY.  [bOOK  II. 

the  axis  of  Y,  therefore,  intersects  the  circumference  in 
two  points,  equally  distant  from  the  origin,  one  above  the 
axis  of  -Zj   and  the  other  below  it. 

3.    To  trace  the   curve    between  these    points,  find  the 
valu6   of  y  from  the  equation  of  the   circle,  which  gives, 


y  =    ±  V-^  -  x\ 

Now,  since  every  value  for  jc,  gives  for  y  two  equal 
values,  with  contrary  signs,  it  follows  that  the  curve  is 
symmetrical  with  respect  to  the  axis  of  SS';  and  in  the 
same  manner,  it  may  be  shown  to  be  symmetrical  with 
respect  to  the   axis  of   Y. 

Beginning  at  the  point  where    x  =  0,    we  have, 

y  =  ±  i?. 

The  values  of  y  then   decrease,  numerically,  as  x  increases 
numerically ;   and  when  x  becomes  equal  to  ±  H,  we  have, 

2/  =  0; 

hence,   the    curve    intersects  the   co-ordinate   axes    in    four 
points,  at  a  distance  from  the   origin,   equal  to  H. 

4.  If  X  becomes  greater  than  ±  i2,  the  values  of  y 
become  imaginary,  which  shows  that  the  curve  is  limited 
both  in  the   direction  of  x  positive,   and  of  x  negative. 

By  placing  the   equation  under  the  form, 


X  =   znV^^-yS 

we  may  show,  that  the  circumference  is  also  limited  in  the 
direction  of  y  pcJsitive,  and  in  that  of  y  negative. 


BOOK   II.]  THE      CIRCLE  63 

5.    By    attributing    a   particular  value  to    either    of  the 
variables,  in  the   equation, 


2/  =   ±V^^ 


the  corrcq)onding  values  of  the  other  variable  may  be  found. 
If  we  suppose    i2  =  1,    and  then  make. 


a;  =  0, 

we  have, 

2/  =  ±  1. 

1 

gives. 

y^±^J\     =i^. 

3 

gives. 

y-^A-V^' 

&c.. 

&c., 

&0. 

6.    If,   in 

the 

equation. 

x^-^y'-. 

=  -R', 

X  and  y  denote  the  co-ordinates  of  a  point  within  the  cir- 
cumference, the  equality  will  be  destroyed,  and  x^  -f  y^, 
will  be  less  than  72^,  and  we  shall  have, 

a;2  +  2/2  —  i22  <  0 ; 
that  is,  negative. 

For  a  point   on  the  curve, 

a.2  +  2/2  _  722  _.  0; 

and  for  a  point  without  the   curve, 

a;2  -f  y2  _  722  >  0 ; 
that  is,  positive. 

7.    The  equation, 

2/2  =  i22  -  x\ 

may  be  put  under  the  form, 

y2  =   {li  +  x)  (H-^ 


64: 


ANALYTICAL      GEOMETRY. 


[book    II. 


in  which  the  factors,  H  +  x,  and  It  —  x,  are  the  two 
segments  into  which  the  ordinate  y  divides  the  diameter; 
this  ordinate  is,  therefore,  a  mean  proportional  between  the 
two  segments.* 

8.  The  equation  of  the  circle  may  also  be  placed  nnder 
another  form,  by  transferring  the  origin  of  co-ordinates, 
fi'om  the  centre  to  a  point  of  the  circumference. 

For  this  transformation,  we  have  the  Formulas  (Bk.  I., 
Art.  2§), 

X  =  a  +  x\    and    y  =  h  ■\-  y\ 

Let   the    origin    be   transferred 
to  B. 

The   co-ordinates   of  this  point 
are, 

a  =   —  B^    and    5  =  0; 
hence, 
a;  =  —  jR  -f  a',     and     y  =  y\ 


Substitutmg  these  values  in  the   equation, 

y-i-   -  JEC-  -  x\ 
we   obtain, 

y'2  =   2Ex'  -  iB'2  ; 

or,   omitting  the  accents, 

2/2  =  2Rx  —  a;2 ; 

which  is  the  equation  of  the  circle  when   the   origin    of  co- 
ordinates is  at  the  left  extremity  of  the  initial  diameter. 

'  9.  When  the  absolute  term  in  the  equation  of  a  line 
is  wanting,  the  line  will  pass  through  the  origin  of  co- 
ordinates. 


Legendre,  Bk.  IV.     Prop.  23.    Cor. 


BOOK  n.J 


THE      CIRCLE. 


65 


For,   the   co-ordinates   of  the   origin  are, 

X  =  0,    and    y  =  0 ; 

these  values  being  substituted  in  any  equation  wanting  the 
absolute  term,  wiU  reduce  both  members  to  0;  hence,  the 
co-ordinates  of  the  origin  will  satisfy  the  equation  of  the 
line;  and,  therefore,  the  line   will  pass  through  the  origin. 

5.     There  is  yet  a  more  general  form  under  which  the 
equation   of  a   circle   may  be   expressed. 

The  characteristic  property  of  the  circumference  of  a 
circle  is,  that  all  its  points  are  at  uu  equal  distance  from 
the  centre.  To  express  this  pro- 
perty, ajialytically^  and  in  a  gen- 
eral manner,  designate  the  co-or- 
dinates of  the  centre  by  x'  and 
y' ;  the  co-ordmates  of  any  point 
of  the  circumference,  by  x  and 
y,  and  the  radius  by  M. 

The   distance    from    any  point, 
whose  co-ordinates  are  x\  y\  to  a  point  whose  co-ordinatca\ 
are  x  and  y  (Bk.  I.,  Art.  19),  is, 

{x  -  xy  +  (y  -  y'Y  =  R\ 

This,  therefore,  is  the  most 
general  equation  of  the  circle  re- 
ferred to  rectangular  co-ordinates. 
By  attributing  proper  values  and 
signs  to  x'  and  y\  the  centre  may 
be  placed  at  any  point  in  the 
plane   of  the  co-ordinate  axes. 

1.    To  find  the  points  in  which 


66 


ANALYTICAL      GEOMETRY. 


[book  II. 


the  circumference  intersects   the   axis   of  Jl,   make    y  =  0, 
and  we  Lave, 


x'  ±  yJi^ 


y 


from  which  we  see,  that  the  values 
of  X  will  become  imaginary  when 
y'  exceeds  i?,  and  it  is  plain  that 
in  that  case  there  will  be  no  in- 
tersection. 


2.    To  find  the  points  in  which  the  circumference   inter- 
sects the  axis  of  I^  make    a;  =  0,  and  we  have, 


y  =  y'  ±  V^^2  _  r^'2^ 

in  which  the  values  of  y  wiU  be  imaginary,  if  x'  exceeds  M, 

3.    If   the    co-ordinates   of  the  F 

centre  of  a  cu'cle  are, 


x'  =   —  2,     and    y'  =   —  4, 

and    the    radius    equal    to    6,    its 
equation  will  be, 


{x  -f  2)2  +  (y  -f  4)' 


36, 


N 

' 

/ 

A 

V 

c~ 

y 

1 

from  which  the  cii'cumference  may  be  readily  described. 
Find  the  points  in  which  it  cuts  the  co-ordinate  axes. 

Supplementary  chords. 

6.  ScTPLEMENTARY  Chokds,  are  pairs  of  chords  drawn 
through  the  extremities  of  a  diameter,  and  intersecting 
each  other  on  the   curve.  / 


BOOK  n.] 


THE      CIRCLE. 


67 


Supplementary  chords  of  the  circle  are  at  right  angles. 

7.    Let  A  be  the  origin  of  co-ordinates,  and  JB  and  JB\ 

the  extremities  of  a  diameter. 

The  equation  of  a  straight  line 
passing  through  a  given  point,  is 
of  the  form  (Bk.  I.,  Art.  20), 


y  —  ^'  =  a{x  -  x'). 

If  the  line  passes  through  the 
point  iS,  whose  co-ordinates  are 
£c'  =  +  i?,  and  y'=  0,  its  equa- 
tion will  be, 

y  =  a(x  —  72) 


^.Y 


(1-) 


For  a  like  reason,  the  equation  of  a  straight  line  passing 
through  i?',  whose  co-ordinates  are  x'  =  —  JR,  and  y'  =  0,  is, 

y  =  aXx  +  H) (2.) 

If  these  two  lines  intersect  each  other,  the  co-ordinates 
of  their  point  of  intersection  will  satisfy  both  equations. 
Hence,  if  we  suppose  x,  in  one  equation,  to  be  equal  to  x 
in  the  other,  and  y  equal  to  y,  and  then  combine  the 
equations  by  multiplying  them  together,  member  by  mem- 
ber, the  resulting  equation, 

2/2  =  aa'ix'  ■-  E-'),    ....    (3.) 

will  express  the  condition,  that  the  two  straight  lines  shall 
intersect  on  the  plane  of  the  co-ordinate  axes. 

But,  if  the  point  of  intersection  is  in  the  circumference  of 
the  circle,  a  and  y  must  satisfy  the  equation. 


a.2  4.  2/2 


M\ 


(4.) 


68  ANALYTICAL      GEOMETRY.  [bOOK  n. 

or,  y2  ^  i22  _  2,2  ^   _  i(a;2  _  i22). 

Hence,  aa'  =  —  1,    or,    aa'  +  1  =  0. 

The  two  supplementary  chords,  therefore,  are  at  right 
angles  (Bk.  I.,  Art.  23—2). 

1.  In  the  equation  of  condition, 

aa'  -\-  I  =  0, 

the  two  tangents  a  and  a',  are  undetermined;  there  are, 
therefore,  an  infinite  number  of  values  which  may  be  at- 
tributed to  either  of  them,  that  will  satisfy  the  equation  ; 
hence,  there  is  an  indefinite  number  of  supplementary  chords 
that  may  be  drawn  through  the  extremities  of  the  same 
diameter,  each  pair  of  which  will  be  at  right  angles. 

2.  If  one  of  the  supplementary  chords  makes  a  given 
angle  with  the  axis  of  -Z",  its  tangent,  a  or  a',  is  known ; 
and  tlien,  the  value  of  the  other  tangent  may  be  found  from 
the  equation  of  condition. 

3.  If  either  a  or  a'  is  equal  to  0,  the  other  will  be  in- 
finite ;  which  shows,  that  if  one  of  the  chords  coincides  with 
the  axis  of  -X^,  the  other  will  be  perpendicular  to  it. 

Tangent  line  to  the  circle. 

8.  Two  points  of  a  curve  are  consecutive^  when  there  is 
no  point  of  the  curve  between  them. 

A  straight  line  is  tangent  to  a  curve  when  it  has  two 
consecutive  points  in  common  with  the  curve. 

To  find  the  equation  of  a  tangent  line  to  a  circle. 

Let  A  be  the  origin,  and  the  equation  of  the  circle, 

x^  +i/^=Ji^, (1.) 


BOOK  II.] 


THE      CIRCLE. 


69 


Take  any  point  of  the 
circumference,  as  P,  and  de- 
signate its  co-ordinates  by 
x'\  y".  Through  this  point 
draw  a  secant  line ;  its 
equation  will  be  of  the  form 
(Bk.  I.,  Art.  20), 


y  -  y"  =  a(x  -  x") 


(2.) 


It  is  required  to  find  the  value  of  a,  when  the  secant 
'line  PB  becomes  tangent  to  the  circumference. 

Since  the  point  P  is  in  the  circumference,  its  co-ordinates 
will  satisfy  the  equation  of  the  circle,  and  we  shall  have, 

JB"2  4.  y"2  ^722 (3) 

Subtracting  Equation  ( 3 )  from  ( 1 ),  member  from  mem- 
Ijer,  we  obtain. 


or,      {x  4-  a;")  {x  -  cc")  -f  (y  -f  y")  (y  -  y") 


(4.) 


in  which    equation,   x  and    y  are  the  co-ordinates  of  any 
point   of  the    circumference. 

If  Equation  (4)  be  combined  with  (2),  a;  and  y,  in  the 
resulting  equation,  will  be  the  co-ordinates  of  the  points  in 
which  the  secant  intersects  the  circumference.  The  equa- 
tions are  most  readily  combined,  by  substituting  for  y  —  y", 
in  Equation  (4),  the  value  of  y  —  y"  in  Equation  (2). 
Making  the   substitution,   we   obtain, 

(X  +  a;")  {x  -  JK'O  -f  (y  +  y")  a  (aj  -  x")  =  0, 

and,  by  fiictoring,  we  have, 


70  ANALYTICAL      GEOMETRY.  [bOOK  H. 

{x  -  x")  X  [ic  +  x"  4-  a[y  +  y")]  =  0, 

which  is  satisfied  by  making, 

x  —  x"  =  0,      or,      a;  +  aj"  +  a{y  +  y")  =  0. 

In  the  first  equation,  x  denotes  the  abscissa  of  the  point 
P;  in  the  second,  of  P'. 

If  we  suppose  the  secant 
PP  to    turn    around    the 
point     P,    the     point     P 
will     approach     P  ;      and 
when  P'  shall  become   con- 
secutive with  P,  the  secant 
line  will  become  tangent  to 
the     circumference.      When 
this  takes  place,  we  shall  have, 

X  z=  x'\      and      y  =  y'\ 

and  the  second  equation  will  give, 


a   = r,, 

y 


a;",   y'\  being  the  co-ordinates  of  the  point  of  contact. 

ion  (2),  we 


Substituting  this  value  in  Equation  (2),  we  have, 

y-y  =-y. 

or,  by  reducing, 

yy"  _  2/"2  _   _  a-'^a;  ^  aj'/a^ 


or, 
or, 


yy"  +  xx"  =  2/"2  +  aj 


BOOK    n.]  THE      CIRCLE.  71 

in  which   X  and  y  are  the  general  co-ordinates  of  the  tan- 
gent line. 

1.  For  the  point  in  which  the  tangent  intersects  the  axis 
of  J^  we  have  a;  =  0,   and, 

if  yn 

2.  For  the  point  in  which  the  tangent  intersects  the  axia 
of  X^  we  have,  y  =  0^  and, 

a;  =  ^,  =  AT. 

x" 


Normal  line. 

9.  A  Normal  Line  to  a  curve,  is  a  line  perpendicular 
to  the  tangent    at  the  point  of  contact. 

Every  normal  line,  in  a  circle,  passes  through  the  centre. 

10.  The   tangent  of  the   angle   which  the  tangent   line 
to  a  circle  makes  with  the  axis  of  JT  (Art.  8),  is. 

The   equation   of   any   straight    line   passing  through  the 
point  of  tangency  will  be  of  the  form, 

y  -  y"  =  «'(«  -  ^")' 

The  equation  of  condition  requiring  this  line  to  be  perpen- 
dicular to  the  tangent  (Bk.  I.,  Art.  23 — 2),  is, 

aa'  -f  1   =  0,         or,         a'  = .     .      (2.) 


72  ANALYTICAL      GEOMETRY.  [bOOK   II. 

Substituting  for  a  its  value  in  Equation  ( 1 ),  we  have, 
a'=K,. 
The  equation  of  the  normal,  therefore,  becomes, 

y  -  y"  =  J;'(«  -  ^"), 

or,  by  rtducmg, 

yx"  -y"x  =  0;  or,  y  =  ^,x', 
and  since  this  equation  has  no  absolute  term  (Art.  4 — 9), 
the  line  passes  through  the  origin  of  co-ordinates.  We  have 
thus  proved  a  property  well  known  in  Elementary  Geometry, 
viz. :  that  a  line  perpendicular  to  the  tangent,  at  the  point 
of  contact,  passes  through  the  centre  of  the  cii'cle. 

Polar  equation. 

11,  The  polar  equation  of  a  curve,  is  the  equation  which 
is  obtained  by  referring  the  curve  to  a  fix^d  point  and  a 
given  straight  hne.  The  fixed  point  is  called  the  pole ;  the 
variable  distance,  from  the  pole  to  any  point  of  the  curve, 
is  called  the  radius-vector ;  and  the  angle  which  the  radius- 
vector  makes  with  the  given  straight  line,  is  called  the  vari- 
able  angle. 

Polar  equation  of  the  circle. 

12.  Let  it  be  required  to  find  the  polar  equation  of  the 
circle,  when  the  pole  is  in  the  circumference. 

The  equation  of  the  circle,  referred  to  rectangular  co-or- 
dinates, when  the  origin  is  in  the  circumference,  as  at  £\ 

(Art.  4—8),  is, 

y'^  z=  2Rx  ^  x^    ,    ^    ,     ,    .     (1.) 


BOOK  n.] 


THE      CIRCLE. 


73 


If  B'  is  the  pole  of  the 
polar  co-ordinates,  we  have 
(Bk.  I.,  Art.  36—1), 

a;=rcosv,    and    y^zrsinv.        _ 

B 

Substituting  these  values 
of  X  and  y,  in  Equation  ( 1 ), 
we  have, 

r^  sin^v  =  2i?r  cos  v  —  r^  co^v. 
Transposing,  and  remembering  that, 

sin^y  -f  cos^y  =  1, 


we  have. 


2Er 


cos?; 


0; 


which  is  the  polar  equation  of  the  circle  when  the  pole  is 
at  B\  and  the  angle  v^  estimated  from  the  axis  of  -Xl 


Interpretation  of  the  equation. 

13.     Since  the  polar  equation, 

r^  —  27?r  cos  v  =  0, 

has   no   absolute  term,  one   of  the   roots   is   equal   to    0 ;  * 
which  ought  to  be  the  case,  since  the  pole  is  on  the  curve 
(Art.  4—9). 
'  Dividing  by  this  value  of  r,  we  obtain  for  the  other  value, 

r  =  272  cos  V. 

This  value  of  r  will  be  positive,  when  the  cosv  is  posi. 
tive  ;   and  negative,  when  the   cos  v  is  negative.    But  the 
*  Bourdon,  Art.  251.     University,  Art.  193. 


74  ANALYTICAL      GEOMETRY.  [bOOK   U. 

negative  values  of  the  radius-vector  must  be  rejected,  since 
they  cannot  enter  into  the  analysis  (Bk.  I.,  Art.  35). 

The  figure  also  indicates  the  same  result.  For,  the  cosw 
is  positive  in  the  first  and  fourth  quadrants  ;  hence,  the 
radius-vector  is  positive  when  it  falls  in  the  first  or  fourth 
angle.  The  cosr,  is  negative  in  the  second  and  third 
quadrants ;  hence,  the  radius-vector  is  negative  when  it 
fells  in  the   second  or  third  angle. 

For  V  =  0,  the   cos  u  =  1,   and  we  have, 
r  =  2R  =  JB'B. 

When  V  increases  from  0  to  90°,  the  radius-vector  con- 
tinues positive,  and  determines  all  the  points  in  the  semi- 
circumference  BCJB'. 

Tliis  may  also  be  verified.  For,  in  the  right-angled 
triangle  B' CB, 

B'C  z=  B'B  cos BB'O;* 

that  is,  r  =  2i2  cos  v. 

When  V  becomes  equal  to  90°,  cosv  =  0,  and  r  is  0. 
The  radius-vector  then  becomes  tangent  to  the  circumfer- 
ence, since  the  two  points  in  which  it  before  cut  it,  have 
united. 

From  V  =  90°,  to  t?  =  270°,  the  cosy  is  negative; 
and  there  is  no  point  of  the  curve  either  in  the  second 
or  third  angle. 

From    V  =  270°,    to    v  =  360°,   the    cos  v    is    positive, 
and  the  radius-Vector  will   determine  all  the  points  of  the 
semi-cu'cumference,   below  the   axis  of  abscissas. 
*  Trig.  Art.  37. 


BOOK  n.] 


THE      CIRCLE, 


75 


2.  If  the  pole  be  placed  at  the  point  7?,  whose  co-ordi- 
nates are, 

a  =  +  By        6  =  0, 

the   equation  will  become, 

r  =   —  2R  cosw. 

In  this  equation  the  ra- 
dius-vector will  be  negative, 
when  cos  v  is  positive,  and 
positive,  when  the  cost;  is 
negative. 

Hence,  the  radius-vector 
will  not  give  points  of  the 
curve  from  v  =  0,  to  v  =  90°.  It  will  give  points  of  the 
curve  from  v  =  90°,  to  v  =  270° ;  and  it  Avill  again  fail  to 
determine  a  curve  from  v  =  270°,  to  v  =  360^.  The  ficj- 
ure  verifies  these   results. 

3.  If  we    place  the    pole   at   the    centre,  the    equations 
for  transformation,  will  become, 


r  cos  Vy 


y  =  r  sm  0. 


BOOK    III. 


D  p 


PD' 


OF      THE      ELLIPSE. 

1.  An  Ellipse  is  a  plane  curve,  such,  that  the  sum  of 
the  two  distances  from  any  point  of  it,  to  two  fixed  points, 
is  equal  to  a  given  distance. 
Thus,  if  F  and  i^be  two  fixed 
points,  and  AB  a  given  dis- 
tance;  then,  if  i^P  +  Pi^,  is 
constantly  equal  to  AB^  for 
every  position  of  the  point  P, 
the  curve  APBP  will  be  an 
ellipse. 

1.  The  fixed  points,  J?"  and  F,  are  called  foci  of  the 
ellipse. 

2.  The  line  AB^  passing  through  the  foci,  and  limited 
by  the  curve,  is  called  the  transverse  axis;  and  the 
extremities  A   and  P,  the  vertices   of  the  transverse  axis. 

3.  The  point  C,  on  the  transverse  axis,  and  equally  dis- 
tant from  the  foci  F'  and  F^  is  called  the  ceiitre  of  the 
ellipse. 

4.  The  line  DD'  drawn  through  the  centre,  perpendi- 
cular to  the  transverse  axis,  and  limited  by  the  curve, 
is  called  the  conjugate  axis^  and  I>  and  B'  are  its  vertices. 


Construction  of  the  Ellipse. 
2. — 1.    We  can  easily  cc»nstruct  an  ellipse  when  its  trans- 
verse   axis  and  foci  are  given. 


BOOK  III.] 


THE      ELLIPSE, 


77 


Let  F'  and  F  be  the  foci, 
and  AB  the  transverse  axis. 
Take  a  thread,  equal  in  length 
to  AB^  and  fasten  its  two  ex- 
tremities, the  one  at  F\  and 
the  other  at  F.  Press  a  pencil 
against  the  thread,  and  move 

it  around  the  points  F^  F\  keeping  the  thread  constantly 
stretched :  the  point  of  the  pencil  will  describe  an  ellipse ; 
for,  in  every  position  of  the   pencil,  we  shall  have, 

F'P  ■{-  PF  =  AB, 

which  is  the   characteristic  property  of  the   curve. 

2.    When  the  pencil  is  at  j5,   we  have, 
AB  =  BF'  -f  BF\    but,    BF'  =  FF'  +  FB\    hence, 

AB  =  FF  +  2BF   ....     (1.) 
When  the  pencil  is  at  A,  we  have, 

AB  =  AF+  AF;    but,    AF  =  FF'  +  AF' ;    hence, 
AB  =  FF'+2AF    ....    (2.) 

Equating  the  second  members  of  Equations  (1)  and  (2), 
cancelling  the  common  term  FF\  and  dividing  by  2,  we 
have,  BF=AF; 

tKat  is,  the  distance  from  either  focus  to  the  nearest  vertex 
of  the  transverse  axis  is  equal  to  the  distance  from  the 
other  focus  to  the  other  vertex. 

Since  the  centre  G  is  the  middle  point  of  FF{Kvi.  1-3), 
it  follows,  that  the  centre  C  is  also  the  middle  2^oi?it  of 
the  transverse  axis. 


78 


ANALYTICAL      GEOMETRY.  [bOOK  HI. 


3.  We  may  also  construct  the  ellipse  by  points^  when 
the  transverse  axis  and  foci  are  given. 

Let  AB  be  the  transverse 
axis  of  an  ellipse,  andi^'  and 
F  the  foci.  Take,  in  the  di- 
viders,  any  portion  of  the  trans- 
verse axis,  as  AD^  and  with 
the  focus  F\  as  a  centre,  de- 
scribe the  arcs  p  and  q.     With 

JBD,  the  remaining  part  of  the  transverse  axis,  as  a  radius, 
and  the  other  focus  F,  2i^  2i.  centre,  describe  two  other 
arcs  intersecting  the  former;  the  points  of  intersection  will 
be  points  of  the  curve.  For,  the  sum  of ,  the  distances 
from  p  or  q^  to  F'  and  F^  is  equal  to  AB. 

If  with  the  radius  AJD^  two  arcs  be  described  from  the 
focus  jP,  and  with  the  radius  BD  two  arcs  be  described 
from  the  focus  F\  these  arcs  will  also  determine,  by  their 
intersections,  two  points  of  the  curve;  so  that,  for  each 
time  we  take  a  part  AD  of  the  transverse  axis,  we  shall 
determine  four  points  of  the   curve. 

4.  Construct  an  ellipse  when  its  axes  are  given. 
If  from  either  vertex  of  the 

conjugate  axis,  as  i>,  the  lines 
DF\  DF^  be  drawn  to  the 
foci,  they  will  be  equal  to  each 
other. 

For,  in  the  two  right-angled 
triangles,  F' CD,  FCD,    OF' 

is  equal  to  CF,  and  CD  is  common;  hence,  the  hypo- 
thenuse   DF'  is  equal  to   DFJ^ 

»  Legendre,  Bk.  1.      Prop.    V. 


.       D 

^■\ 

c    yrf 

BOOK  in.] 


THE      ELLIPSE. 


79 


But  F'D  -I-  DF  is  equal  to  AB\  hence,  DF\  or  JDF, 
is  equal  to  OB.  If,  therefore,  with  either  vertex  of  the 
conjugate  axis  as  a  centre,  and  with  a  radius  equal  to 
half  the  transverse  axis,  the  circumference  of  a  circle  be 
described,  it  mil  intersect  the  transverse  axis  at  the  foci. 
Having  found  the  foci,  the  ellipse  may  be  constructed,  as 
in  the  last  case. 

Equation  of  the  ellipse. 


3.  Let  F'  and  F  be  the  foci, 
and  denote  the  distance  between 
them  by  2c ;  then,  GF  or  CF' 
=  c. 

Let  F  be  any  point  of  the 
curve.  Designate  the  distance 
FF,  by  r\  and  FF,  by  r.     Let 

2^  =  AB,   denote  the  the  given  line,  to  which  the  sum, 
F'F  +  FF,   is  to  be  constantly  equal. 

Through  C,  the  middle  point  of  F'F,  draw  OB  perpen- 
dicular to  F'F,  and  let  C  be  the  origin  of  a  system  of  rect- 
angular co-ordinates,  of  which  AB,  I)D\  are  the  axes.  De- 
note the  distance  CB  by  B^  and  let  x  and  y  denote  the 
co-ordinates  of  any  point,  as  P. 

The  square  of  the  distance  between  any  two  points,  of 
which  the  co-ordinates  are  jc,  y,  and  x\  y'  (Bk.  I.,  Art. 
19),  is, 

(y  -  y'Y  +  (aJ  -  cc')2. 

If  the  distance  be  estimated  from  the  point  F\  of  which 
the  co-ordinates  are  y'  =  0,  and  x'  =  —  c,  we  shall  have, 


80  ANALYTICAL      GEOMETRY.  [bOOK   ITI. 


F'F^  =  r'2  =z  y'^+  {x  +  cy    .     .     .     (1.) 

K  it  be  estimated  from  the  point  I\  of  wHch  the  co-ordi- 
nates are  y'  =  0,   and  x'  =  -i-  Cy  we  shall  have, 

r^=  y^+  (x-cy      ....   .(2.) 

Since  the  lines  intersect  each  other,  the  co-ordinates  of 
P  will  satisfy  Equations  ( 1 )  and  ( 2 ) ;  hence,  the  equations 
are  simultaneous. 

If  we  add  and  subtract  them,  we  obtain, 

r'2  -f-  7-2  =    2(2/2  +  a;2  +  ^)     .      ,      ,       (3,) 

and  r'2-  r2  =  4ciB (4.) 

Equation  (4)  may  be  placed  under  the  form, 

(r'-f  r)  {r'  -  r)  =  4cx      ...     (5.) 
But  we  have,  from  the  property  of  the  ellipse, 

r'  +  r  =  2A (6.) 

Combining  (5)  and  (6),  we  have, 

^  -^  =  -3- O') 

Combining  (6)  and  (7),  by  addition  and  subtraction,  we 
obtain, 

r'=  A  +  ^.   .  (8.)        and        r  =  A-^   .  .  (9.) 

Squaring  both  members  of  Equations  (8)  and  (9),  com- 
bining  the  resulting  equations,  and  substituting  the  values 
of  r'2  and  r-,  in  Equation  (3),  we  obtain. 


^'  +  ^'  =  y'  +  «^'  +  «'• 


BOOK  ni.] 


THE      ELLIPSE. 


81 


Substituting  for  c^  its  value,  A"^  —  JB^,  (Art.  2 — 4),  we 
have, 

A''  +  ^^—j^  052  =  2/2  +  a;2  4.  ^2  _  ^  . 

or,  A^  4-   A^X^  -  i?2a,2   _    ^2y2  ^   ^23.2  4.   ^4   _   J[2^2^ 

Cancelling  and  transposing,  we  have, 

^2y2   +    J52a.2    _    ^2^2^ 

which  is  the  equation  of  the  ellipse,  referred  to  its  centre 
and  axes. 

Interpretation  of  the  equation. 

4.     1.   If,  in  the  equation  of  the  ellipse, 

we  make  y  =  0,  the  corresponding  values  of  x  will  be  the 
abscissas  of  the  points  in  which  the  curve  intersects  the 
axis  of  X  (Bk.  n.,  Art.     4—1),  viz.: 


X  =   -\-  A,     for    J3, 


and 


X  =   —  A,     for    A, 


2.  If  we  make  cc  =r  0,  the  corresponding  values  of  y  will! 
be  the  ordinates  of  the  points  in  which  the  curve  intersects- 
the  axis  of  IT:  viz. : 


y 


+  B,    for    J9,        and 


3.  If  we  place  the  equation  of 
«■ 
the  ellipse  under  the  form, 


y  =  ±  ^V^^  -  x% 

we  see,  that  for  every  value  of 
X,    as     Clly    whether     plus     or 
6 


82 


ANALYTICAL      GEOMETRY, 


[book  hi. 


minus,  there  will  be  two  values  for  y,  numerically  equal, 
with  contrary  signs  ;  hence,  the  curve  is  symmetrical  with 
respect  to  the  transverse  axis. 

If'  X  be  made  greater  than  A^  whether  it  be  taken  plus 
or  minus,  the  values  of  y  will  be  imaginary ;  hence,  the 
curve  will  be  limited  both  in  the  direction  of  x  positive  and 
X  negative. 

4.    If   we   place    the    equation 
under  the  form, 


a;  =   ±  -^^B^ 


y\ 


we  see,  that  for  every  value  of 
y,  whether  positive  or  negative, 

as  CX^  there  will  be  two  equal  values  of  ic,  with  con- 
trary signs ;  hence,  the  curve  will  be  symmetrical  with  re- 
spect to  the  conjugate  axis.  If  y  be  made  greater  than  j5, 
either  positive  or  negative,  the  values  of  x  will  be  imagin- 
ary ;  hence,  the  curve  will  be  limited  in  the  direction  of  y 
positive  and  y  negative. 

5.    The  equation  of  the  ellipse, 

may  be  put  under  the  form, 

and  this  equation  will  be  satisfied, 

80  long  as   X    and    y    denote  the  co-ordinates  of  points  of 

the  curve. 

If  we  take  any  point,  as  P',  without  the  curve,  its  ordi 
nate  JP'Z^,  will  be  greater  than  the  ordinate  of  the  curve 


BOOK  in.] 


THE      ELLIPSE. 


83 


If  we  denote  this  ordinate  by  y,  the  first  member  of  the 
equation,  instead  of  reducing  to  0,  will  be  equal  to  a  posir 
live  quantity. 

If,  on  the  contrary,  we  take  a  point  P",  within  the  curve, 
its  ordinate  F"JD,  will  be  less  than  the  ordinate  of  the 
curve ;  and  if  we  designate  this  line  by  y,  the  first  member 
of  the  last  equation  will  be  negative. 

Therefore,  the  following  analytical  conditions  will  deter- 
mine the  position  of  a  point,  with  respect  to  the  curve  of 
the  ellipse,  viz.: 

Without  the  curve, 
In  the  curve, 
Within  the  curve, 


^2y2  ^   ^23.2  _   ^2^2  ^  0. 
J2y2   ^    J^2rf.2  _    ^2^2   <-    0. 


.^, 


Equation  when  the  origin  is  at  the  vertex  of  the  transverse  axis. 

5.  If  we  transfer  the  origin 
of  co-ordinates  from  the  centre 
(7,  to  A^  one  extremity  of  the 
transverse  axis,  the  equations 
of  transformation  (Bk.  I.,  Art. 
28),  will  reduce  to, 

X  =z  -  A  -{-  x\  y  =  y'- 

Substituting  these  values  in  the   equation  of  the  ellipse, 
it -reduces  to, 

^2y'2  ^  ^2a.'2  _  2B''Ax'  =  0, 

which  may  be  put  under  the  form, 


B 


^ 


y''  =  jjS."^^^'  -  «'")>      or>     y'  =  3-2(2-40;  -  x^) 


84 


ANALYTICAL      GEOMETRY, 


[book   III. 


by  omitting  the  accents.  This  is  the  equation  of  the  ellipse 
refeiTed  to  the  vertex  of  the  transverse  axis,  as  an  origin 
of  co-ordinates.  In  this  equation,  the  absolute  term  is  want- 
ing, as  it  should  be,  since  the  origin  of  co-ordinates  is  in 
the  curve  (Bk.  11.,  Art.  4—9). 


Eccentricity.    Polar  equation. 

6.  The  ECCEXTEicTTY  of  an  ellipse,  is  the  distance  from 
the  centre  to  either  focus,  di\'ided  by  the  semi-transverse 
axis.  If  c  denotes  the  distance  from  the  centre  to  either 
focus  (Art.  3),  and  e  the  eccentricity,  then, 


^/A^^B' 


and      c  =  Ae 


(1.) 


1.    Resuming  Equations  (8)  and  (9)    (Art.  3),  we  have, 
r'  =z  A  +  ex    .    .    (2.)         and         r  =  A  —  ex    .    .    (3.) 

For  the  value  of  r',  the  pole  is 
at  F' ;   for  r,  it  is  at  J^  and  the 
origin    of   co-ordinates    is    at    the  A\ 
centre   of  the   ellipse. 

Let  us  transfer  the  origin  of  co- 
ordinates to  the  focus  F'.  For 
this  point  we  have  (Bk.  I.,  Art.  28), 

X  =   —  Ae  ■\-  x\        and 

Substituting  this  value  of  a;  in  Equation  ( 2 ),  we  have, 

r'  =  A  -  Ae^  +  ex'. 

But  x'  =  r'  cosv  (Bk.  I.,  Art.  35).    Substituting  this  value 
of  x\  we  have, 


y  =  2/ 


BOOK   III.]  THE      ELLIPSE.  85 

r'  =  A  —  Ae"-  +  er'  cos  v  ; 

whence,  r'  =  -—^ -i (4.) 

1  —  e  cos  V 

which  is  the  polar  equation,  when  the  pole  is  at  F\ 

2.  If  ?;  =  0,   cosw  =  1,  and  we  have, 

^,  ^  A{\  -  e^)  ^  ^n  _^  ^)  ^  ^  ^  ^  ^  ^rj^ 
1  —  e 

If  V  increases  from  0  to  360°,  the  corresponding  values 
of  r\  will  give  all  th6  points  of  the  curve. 

3.  K  we  had  transferred  the  origin  of  co-ordinates  to 
the  focus  JFJ  we  should  have  had  (Bk.  I.,  Art.  28), 

X  =  Ae  +  x\        and        y  —  y'. 

Substituting  this  value  of  cc  in  Equation  (3),  we  have, 

r  =  A  —  Ae^  —  ex'. 

But  x'  =  r  cos  V.     Substituting  this  value  of  x\  we  have, 

r  =  -4  —  Ae^  —  er  cos  v ; 

,                                        A(l  -  e^) 
whence,  r  =   — ^^ ~ (5.) 

1  +  e  cos  V  ^    ' 

4.  We  see,  from  Equations  (2)  and  (3),  that  when  the 
pole  is  at  either  focus,  the  radius-vectors  will  be  expressed 
in  ratio7ial  functions  of  the  abscissas  of  the  points  in  which 
they  intersect  the  curve.  It  may  be  easily  shown,  that  the 
foci  are  the  07ily  points  in  the  plane  of  the  curve^  which 
enjoy  this  remarkable  property. 

5.  If  u  =  0,   cosy  =  1,   and  we  have, 

r  =  ^[}  -  ^')  =  A{l-e)  =  A-c  z=  FJB. 
I  +  e  ^ 


86 


ANALYTICAL      GEOMETRY. 


[book  m. 


If  V  varies  from  0  to  360°,  the  corresponding  values  of  r 
will  give  all  the  points  of  the  curve.  The  difference  be- 
tween Equations  (4)  and  (5)  is  this:  for  y  =  0,  the  value 
of  r'  begins  at  the  remote  vertex  j  while  in  Equation  (5), 
under  the  same  supposition,  the  corresponding  value  of  r, 
begins  at  the  nearest  vertex. 


Diameters. 


7,  A  DIAMETER  of  an  ellipse,  is  a  line  drawn  through 
the  centre,  and  limited  by  the  curve.  The  points  in  which 
it'  intersects  the  curve,  are  called  vertices  of  the  diameter. 


Every  diameter  is  bisected  at  the  centre. 

8,    The  equation  of  the  ellipse,  referred  to  its  centre  and 
axes,  is, 

^2^2  _,_    ^2^.2   ^    ^2^2      ....        (1.) 

Since  every  diameter,  as 
II'  CH^  passes  through  the  ori- 
gin of  co-ordinates,  its  equa- 
tion will  be, 

y  =  ax    .    ,     (2.) 

If  Equations  ( 1 )  and  ( 2  )  be 
combined,  the  values  of  x  and  y,  in  the  resulting  equation, 
will  be  the  co-ordinates  of  the  vertices  II  and  H'. 

Combining  and  eliminating,  we  have, 


"  =  ± ^^V3?+^'    y  =  ± ^^« V2S 


+  jp 


BOOK    III.] 


THE      ELLIPSE. 


87 


If  we  denote  the  co-ordinates  of  the  point  11^  by  x\  y\ 
and  the  co-ordinates  of  ZT,  by  x'\  y'\  we  shall  have, 


Since  the  co-ordinates  of  these  points  are  the  same,  with 
contrary  signs,  it  follows  that, 

that  is  :   Every  diameter  is  bisected  at  the  centre, 

Ordinates  to  diameters. 

9.  An  ordinate  of  a  point,  on  a  curve,  to  a  diameter, 
is  its  distance  from  the  diameter,  measured  on  a  line  paral- 
lel to  a  tangent  at  the  vertex.  The  parts  into  which  the 
ordinate  divides  the  diameter,  are  called  seyme?its. 

Relation  of  ordinates  to  each  other. 

10.  The  equation  of  the  el- 
lipse referred  to  the  vertex  A, 
as  the  origin  of  co-ordinates 
(Art.  5),  is. 


2/' 


'  -  J>^ 


X)X. 


If  we  designate  a  particular  ordinate  by  y',  and  its  ab- 
scissa by  x' ;  and  a  second  ordinate  by  y'\  and  its  abscissa 
by  x'\  we  shall  have. 


88 


and, 


ANALYTICAL      GEOMETRY. 


y'^  =  ^(2^  -  x')x',    . 
y'--  =  J(2^  -  x")x"   . 


[book  m. 
.     (1.) 
.     (2.) 


Dividing  Equation  ( 1 )  by  ( 2  ),  we  obtain, 


y" 


(2A  -  x')x'  ^ 
{2 A  -  x")x"' 


or. 


y"2    :  ;     (24  -  x')x'    :     (2A  -  x'')x''. 


But  2A  denotes  the  trans- 
verse axis  A^,  and  since 
x'  =  AD,  2A  -  x'  =  DB;  ^ 
therefore,  {2A  —  x')x\  is  the 
rectangle  of  the  segments  AD, 
DB.  In    like     manner, 

(2JL  —  x")x",  is  the  product  of  the  segments  AE,  EB, 
Since  the  same  may  be  shown  for  the  conjugate  axis,  we 
conclude  that. 

The  squares  of  the  orclinates,  to  either  axis  of  an  el- 
lipse, are  to  each  other  as  the  rectangles  of  the  correspond- 
ing segments  into  which  they  divide  the  axis. 


Parameter. 

11,    The  Paea:meter  of  the  transverse  axis,  is  the  double 
ordinate   passing  through   the  focus. 

To  find  its  value,  let  us  take 
the  Polar  Equation  (5)  (Art.  6),  A    / 

^(1  -  c2) 


VIZ 


r  = 


1  -\-  e  cos  V 


BOOK   III.] 


OF      THE      ELLIPSE. 


80 


If  we  make  v  =  90°,  the  radius-vector  will  be  per- 
pendicular to  the  transverse  axis,  and  r  will  be  equal  to 
the   ordinate.     Under  this  supposition,    cos  v  =  0,    and  we 

Bhall  have, 

r  =  A(l  —  e^). 


In  Art.  6    we  have, 


92     _ 


A^-  JB^ 


Substituting  this  value  for  e\  we   have, 


= '■(■  -  ^^) = 


Hence,        parameter  =  —j-  z=  -^—r  =  2r 


A' 
YA 


A 


If  we  write  this  in  a  proi^ortion,  we  have, 

1A    :    IB    '.'.    2B    :    parameter ;    that  is. 

The  parameter  of  the  transverse  axis  is  a  third  jyro- 
portional  to   the   transverse  axis   and  its  conjugate. 

1.  In  the  polar  equation,  when  the  pole  is  at  the  focue, 
the  numerator^  in  the  value  of  r,  is  equal  to  half  ttic 
parameter, 

'  Ellipse   and  circumscribing  circle. 

12,  If  on  the  transverse  axis 
AB^  the  cii'curaference  of  a 
circle  be  described,  it  is  re- 
quired to  find  the  relation  be-  Al 
tween  any  ordinate  GD^  of  the  \ 
circle,  and  the  ordinate  IID^  of 
the  ellipse,  corresponding  to  the 
same  abscissa  CD. 


90 


ANALYTICAL      GEOMETRY 


[book  ni. 


Let    Y'   denote   any   ordinate   of  the   circle,   and   y'   the 
corresponding    ordinate    of    the 
ellipse,  and  x'  the  common   ab- 
scissa.   We  shall  then  have  (Bk. 
n.  Art.  4—5), 


J-/2 


A-'-  x' 


(1.) 


We  also  hare  from  the  equa- 
tion of  the  ellipse  referred  to 
its  centre  and  axes  (Art.  3), 


^2 


(2.) 


Dividing   Equation   (2)   by  (1),  member   by   member,    we 
have 


Y'2     -    ^2' 


or     Y,  = 


B 

A' 


B\    that  is, 


Tlierefore,  Y'    \    y'    w    A 

Any  ordinate  of  the  circle,  is  to  the  corresponding  ordi- 
nate of  the  ellipse,  as  the  semi-transverse  axis,  to  the  semi- 
conjugate,  , 

1. ,  It  follows,  from  the  above  proportion,  that  every  or- 
dinate of  the  circle  is  greater  than  the  corresponding  ordi- 
nate of  the  ellipse  :  hence,  every  point  of  the  ellipse,  ex- 
cept the  vertices  of  the  transverse  axis,  is  within  the  cir- 
■cumference  of  the  circle  :  therefore,  the  tra}is verse  axis  is 
greater  t/ia?i  any  other  diameter, 

2.  If  -S  is  made  equal  to  A,  the  ellipse  becomes  the 
circumference  of  the  circle  described  on  the  transverse  axis. 


BOOK  in.] 


THE      ELLIPSE. 


91 


Ellipse  and  inscribed  circle. 

13.  If,  on  the  conjugate  axis, 
DD\  the  circumference  of  a 
circle  be  described,  it  is  re- 
quired to  find  the  relation  be- 
tween any  ordinate,  PN^  of  the 
circle,  and  the  ordinate  J/iV, 
of  the  ellipse,  corresponding  to 
the  same  abscissa  CN. 

Denote  the  ordinates,  NP  and  NM^  by  X'  and  x\  and 
designate  CN^  by  y'.    We   shall  then  have, 


X'2    ^     _2J2 


y'\ 


and, 


^•^  =  ^{B^  -  yn) 


hence, 
If  therefore. 


X'2  -  B^' 

x'         A 
''    X'  =  b'^       . 

X'     :     x'    : 

:     B     :     A\    that  is. 

Any  ordinate  of  the  circle  is  to  the  con^esponding  ordi- 
nate of  the  ellipse^  as  the  seini-co72Jugate  axis  is  to  the 
semi-transverse. 

1.  Since  ^  >  i?,  every  ordinate  of  the  ellipse  is  greater 
than  the   corresponding   ordinate   of  the   circle  :   therefore, 

'every  point  of  the  ellipse,  except  the  vertices  of  the  con- 
jugate axis,  is  without  the  circumference  of  the  circle. 
Hence,  the  conjugate  axis  is  less  than  cmy  other  diameter. 

2.  If  ^  is  made  equal  to  J?,  the  elhpse  becomes  the 
circumference  of  the  circle  described  on  the  conjugate  axis. 


02 


ANALYTICAL      GEOMETEY. 


[book  in. 


(1) 


J^  Equation   of  the    Tangent. 

14.  It  is  required  to  find  tlie  equation  of  a  tangent 
line  to  the   ellipse. 

Take  any  point  of  the  curve, 
as  P,  and  designate  its  co- 
ordinates by  jc",  2/".  Through 
this  point,  di-aw  a  secant  line ; 
its    equation  Tsdll   be   of  the 

form, 

y  -y"  ~  a{x  -  x") 

It   is  now  required   to    find   the  value    of   a,  when   the 
secant  line   PP'   becomes  tangent  to  the   curve. 
The   equation  of  the  ellipse  is, 

^2y2  +  ^2^2  ^    J^2JS2    ....      (2.) 

Since  the  point   P  is  in  the   curve,  we   shall  have, 

A^y"^  +  B'~x''^  =  A^J^   .     ...     (3.) 

Subtracting  (3)  from  (2),  we  have, 

A'~{y' -]/"')  +  P-(^'  -  aj"^)   =  0; 
or, 

A^{y^y"){y-y")-^J^'ix-\-x''){x-x")  =  0.   (4.) 

In  this  equation,  x  and  y  are  the  co-ordinates  of  any  point 
of  the   ellipse. 

If  Equation  (4),  be  combined  with  Equation  (1),  the 
co-ordinates  x  and  y,  in  the  resulting  equation,  will  be  the 
co-ordinates  of  the  points  in  which  the  secant  intersects 
the  ellipse.  These  equations  are  most  readily  combined,  by 
substituting  for  y  —  y'\  m  Equation  ( 4 ),  the  value  in 
Equation  ( 1 ).     Substituting,  we  obtain. 


BOOK   ni.]  THE      ELLIPSE.  93 

or,   by  factoring,   we  have, 

{x  -  x")  X  [A^a{y  +  y")  +  B\x  +  x")-]  =  0, 
an  equation,   which  may  be  satisfied  by  making, 

x-x''  =  0;    or,    A''a{y  +  y")  +  J3^{x  +  x")  =  0. 

In   the   first    equation,   x    is  the    abscissa  of  P;   in  the 
second,  x    and  y  are  the  co-ordinates  of  jP\ 

Suppose  P'  to  move  towards 
P ;  when  they  become  consec- 
utive, we  shall  have, 

x  =  cc",     and    y  =  y'' ; 

which  will  give,  from  the  last 
equation, 


a  =   — 


JPx" 

AY' 


Substituting  this  value  in  Equation  (1),  we  have, 


y  -y 

or,  by  reducing. 


A'^yy"  -  AY""  =  -  B'^xx"  +  B'^x''^ ; 

Dr,  A'yy"  +  B^xx"  =  Ay^  +  PV^ ; 

or,  A^yy''  +  B'^xx''  =  A^B\ 

^hich  is  the   equation   of  the  tangent  line,   and  in  which, 
y  and  x  are  the  general  co-ordinates  of  its  points. 


Sub-tangent. 

15.     A  SUB-TANGENT  is  the  projection  of  the  tangent  on 
the  axis  of  abscissas,  or  on  the  axis  of  ordinates;  that  is, 


9^ 


ANALYTICAL      GEOMETEY.  [bOOK   111. 


it  is  the  part  of  either  axis,  from  the  point  of  intersection, 
to  the  foot  of  the  ordinate  through  the  point  of  tangency. 

1.    To  find  the  sub-tangent,  take  the  equation  of  the  tan- 
gent, 

A^yy"  +  B~xx"  =  A^B^ 

If,    in    this  equation,   we 
make  y  =  0,   we  find. 


X  = 


A^ 


which  is  the  line   OT. 

If  from  CT,  we  subtract 
CHy  which  is  designated 
hj  x",  we  shall  have  the  sub-tangent, 

x'  x" 

2.  This  expression  for  the  sub-tangent  TR^  is  independent 
of  the  conjugate  axis,  and  will,  therefore,  be  the  same  for 
all  ellipses  having  the  same  transverse  axis  AB^  and  the 
points  of  tangency  in  the  same  perpendicular  BP.  Hence, 
if  the  circumference  of  a  circle  be  described  on  the  trans- 
verse axis,  and  the  ordinate  RP  be  produced  till  it  meets 
the  curve  at  §,  the  tangent,  at  this  point,  will  pass  through 
the  common  point  T.  The  angle  CPT^  formed  by  the 
tangent  line  and  the  diameter  passing  through  the  point  of 
contact,  will  be  obtuse. 

3.  If  we  determine,  in  like  manner,  the  sub-tangent  on 
the  conjugate  axis,  it  will  be  independent  of  the  transverse. 

Equation  of  the  normaL 
16.   Since  the   normal   passes  through   the   point   of  tan- 
gency (Bk.  II.,  Art.  9),  its  equation  will  be  of  the  form, 


THE      ELLIPSE, 


95 


BOOK  m.] 

y-y"=  a\x-x"),        .     .     .     (1.) 
and  since  it  is  perpendicular  to  the  tangent,  we  sliall  have, 
aa'  -h  I   =  0, 
But  we  have  found  (Art.  14), 


Ay ' 


hence, 


a' 


^2    x" 


Substituting  this  value  in  Equation  (1),  we  have, 

A^  v" 

y-y"  =  %  |-,(a:  - «"), 

which  is  the  equation  of  the  normal  Hne. 


Sub-normal. 

17.  A  sub-nok:mal  is  the  projection  of  the  normal  on 
the  axis ;  that  is,  it  is  the  part  of  the  axis  which  lies  di- 
rectly under  the  normal. 

1.  To  find  the  sub-normal  AT?,  take  the  equation  of 
the  normal, 

A^  v" 

y-y"  =  ^  h^'^  -  '^")- 

and  make  y  =   0  ;  this  will  give, 

X  =    CJSr  =  ^i!-^— V'  =  e'^x'^     (Art.  6). 

If  we  subtract  this  value  from  (7i?,  which  is  denoted  by 
a?",  we  shall  have  the  sub-normal. 


JV7?  = 


J?2 


A^ 


96 


ANALYTICAL      GEOMETRY. 


[book  in. 


Normal  bisects  the  angle  of  the  two  lines  drawn  to  the  focL 

18.  If  from  P,  any  point 
of  the  curve,  we  draw  two 
lines  to  the  foci  F'  and  F^ 
and  recollect  that  CF\  or 
CF^  is  equal  to  c  =  Ae 
(Art. -6),  we  have,  by  nsing 
the  value  of   CN  -  e^x"   (Art.  17), 

F'K  =  F'G  ^-  CN  =  Ae  +  e'x"  =  e[A  +  ex") ; 
and,  FN  =  CF  -  CN  =  Ae-  e^x"  =  e{A  -  ex"). 
Hence,       FJ^   :    FN   :  :    A  +  ex"    :    A-  ex". 

By  referring  to  the  values  of  r'  and  r  (Art.  3,  Equations 
(8)  and  (9),  and  recollectmg  that  -j  =  e,   "we  have, 

r'    :    r    :  :    A  -{-  ex    :    A  —  €x\ 
hence,  r'    :    r    :  :    F'2T   :    FN; 

therefore,  FN  bisects  the  angle  FFF,* 


Tangent  line  and  lines  to  the  focL 

19.  Let  C  be  the  cen- 
tre of  the  ellipse,  FT  a 
tangent,  and  FF,  FF, 
two  lines  drawn  to  the 
IbcL  Draw  the  normal, 
FN.     Then,    since    NFJI 

and  NPT  are  right   angles,   they  are   equal.     From   each, 
♦Legendre,  Bk.  IV.    Prop.  17. 


BOOK   III.] 


THE      ELLIPSE. 


97 


take  the   equal    angles,   NPF'   and   JSfPF,   and  there  T\all 
remain  F'PIT,  equal  to  FPT.    Hence, 

If  a  line  he  drawn  tangent  to  an  ellipse  at  any  pointy 
and  two  lines  he  drawn  from  the  same  point  to  the  two 
foci-i  the  lines  drawn  to  the  foci  will  make  equal  angles 
with  the  tangent. 

Supplementary  chords. 

20.  Let  AP  be  the  trans- 
verse axis  of  an  ellipse. 

If  a  straight  line  be  drawn 
through  the  point  A^  whose 
co-ordinates  are, 

x'  =  -  A,       y'  =  0, 

its  equation  wiU.be, 

y  =  a'{x  -f-  A). 

If  a  line  be  drawn  through   P,   whose  co-ordinates  are^ 

x'  =  A,         and         y'  =  0, 

its  equation  will  be, 

y  =    a{x  -  A). 

If  these  lines  intersect  each  other,  we  have, 

2/2  _  aa\x^-  J.2);   .     .     .     .     (1.) 

and  if  they  intersect  on  the  curve  of  the  ellipse,  x  and  y 
must  satisfy  the  equation. 


y' 


p^ 


(.12  _x'')    =    -  -'{x''^  -  ^P).    .     .     (2.) 


98  ANALYTICAL       GEOMETRY.  [bOOK   IIL 

By  combining  Equations  (1)  and  (2),  we  have, 

aa   =   —  —  ;     that  is, 

If^  through  the  vertices  of  the  transverse  axis^  two  sup- 
jilementary  chords  he  draicn^  the  iy)^oduct  of  the  tangents 
oj  the  angles  lohich  they  form  with  it,  icill  he  negative^ 
and    equal    to   the  square   of   the  ratio   of   the  semi-axes, 

1.  Since  the  product  of  the  tangents  is  negative,  the 
angles  to  which  they  correspond  will  fall  in  diiferent  quad- 
rants.* 

2.  In  the   equation, 

aa  =  -^,, 

there  are  two  undetermined  quantities,  a  and  cc' ;  hence, 
an  infinite  number  of  pairs  of  supplementary  chords  may 
be   drawn  through   the   extremities  of  the   diameter  A^. 

If,  however,  a  value  be  assigned  to  «,  or  a',  that  is,  if 
one  of  the  supplementary  chords  be  given  in  position,  the 
equation  of  condition  will  determme  the  other,  and  thei'e- 
fore,  the  corresponding  supplementary  chord  may  also  be 
drawn. 

3.  If  the  ellipse  becomes  a  circle,  we  shall  have, 

aa'  =   —  1, 

or,  aa'  -|-  1   =  0  ; 

which  shows,  that  the  supplementary  chords  are  perpen- 
dicular to  each  other,  a  property  bofore  proved  (Bk.  II., 
Art.    7). 

*  Legendre,  Trig.  Art.  59. 


BOOK  III.] 


THE      ELLIPSE. 


99 


Supplementary  chords.    Tangent  and  diameter. 

21.  Let  PT  be  a  tan- 
gent line  to  the  ellipse,  and 
denote  the  co-ordinates  of 
the  point  of  contact  by  a;", 
y".    Then  (Art  14), 

If  a  diameter  be  drawn  through  the  point  P,  its  equation 
will  be  satisfied  for  the  co-ordinates  a;"  y'\  giving, 

and 


2/"  =  a'a", 


x' 


(2.) 


Multiplying  Equations  ( 1 )  and  (  2  ),  member  by  member, 

-52 


aa'  =  — 


A^ 


When  a  and  a'  denoted  the  tangents  of  the  angles  which 
the  supplementary  chords  make  with  the  transverse  axis, 
we  had  (Art.  20), 


hence. 

If,  in  this  equation,  we  make, 

a  =  a, 

we  shall  have, 


a 


a';      that  is,, 


If  owe  chord  is  parallel  to  the  tangent,  the  other  will  be 
parallel  to  the  diameter  passing  through  the  point  of  conr 
tact. 


100  ANALYTICAL      GEOMETKY.  [bOOK   HI. 

Or,  if  we  make, 
we  shall  have, 


a 


a ;      that  is, 


If  one  of  the  chords  he  made  parallel  to  the  diameter^ 
the  other  icill  he  parallel  to  the  tangent, 

1.    Since  the  co-ordinates  of  the  points  P  and  P',  are  the 

same  with  contrary  signs  (Art.  8),  the  value  of  «,  in  Equa- 
tion ( 1 ),  will  be  the  same,  whether  we  consider  the  tangent 
at  P  or  P' ;  hence. 

The  tangents  drawn  through  the  extremities  of  the  same 
diameter  are  parallel. 


y 


Construction  of  tangent  lines  to  the  ellipse. 


22.     Construct  a  tangent  Hne  to  an   ellipse,  at  a  given 
point  of  the  curve,  when  the  axes  are  given. 


First  Method. 

Let  P  be  the  given  point. 
On  the  transverse  axis  ^1j5, 
describe  a  semi-circumfer- 
ence, and  through  P,  draw 
PjR  perpendicular  to  AB^ 
and  produce  it  till  it  meets 
the  circumference  at  P. 
Through  P',  draw  a  tan- 
gent line  to  the  circumference  of  the  circle,  and  from  7J 
where  it  meets  AJB  produced,  draw  7P,  and  it  will  be 
tangent  to  the  ellipse  at  P  (Ait.  15). 

1.    The  angle   GPTh^mg  aright  angle,  the  angle   OPT, 
which  lies  withm  it,  is  obtuse.    Hence,  the  angle  formed 


BOOK  III.] 


THE      ELLIPSE. 


101 


by   a  tangent   line,  and  the   diameter  passing  through   the 
point  of  contact,  is,  in  general,  obtuse. 

If  the  point  of  tangency  be  at  either  vertex  of  the  trans- 
verse axis,  the  tangent  line  to  the  ellipse  will  coincide  with 
the  tangent  line  to  the  circle,  and  will  then  be  perpendic- 
ular to  the  transverse  axis.  Or,  if  the  point  of  contact  is 
at  either  vertex  of  the  conjugate  axis,  the  tangent  line  to 
the  ellipse  will  become  parallel  to  the  tangent  line  to  the 
circle,  and,  consequently,  perpendicular  to  the  conjugate  axis. 

Second  method. 

23.  Let  C  be  the  centre 
of  an  ellipse,  AB  the  trans- 
verse axis,  and  P  the  point 
of  the  curve  at  which  the 
tangent  is  to  be  drawn. 

Through  P,  draw  the 
semi-diameter  P(7,  and  through  A,  draw  the  supplementary 
chord  AII^  parallel  to  it.  Then  draw  the  other  supplement- 
ary chord  JBII^  and  through  P,  draw  FT  parallel  to  BH\ 
then  will  FI"  be  the  tangent  required  (Art.  21). 


Third  method. 

24.  Let  P  be  the  given  point.  Find  the  foci  F'  and  F. 
(Art.  2—4.)  From  P,  draw 
the  lines  FF'  and  FF  to 
the  foci.  Produce  F'F^  un- 
til FM  shall  be  equal  to 
FF,  and  draw  F3L  Then 
draw  FT  perpendicular  to 
FM^  and  it  ^\\\\  be  the  tangent  required,  fiinc^  it  makes 
equal  angles  with  the  lines  FF'  and  FF  [Xvi.  la^. 


102 


ANALYTICAL      GEOMETRY. 


[book  m. 


To  draw  a  tangent  parallel  to   a  given  line. 

25.  Let  AB  he  the  transverse  axis,  and  M  the  given 
line. 

Through  the  vertex  JB 
draw  the  supplementary 
chord  £G,  parallel  to  Jtf. 

Then  draw  AG,  and 
through  the  centre  C  draw 
CF  parallel  to  AG,  and 
produce  it  till  it  meets 
the    ellipse    again     at    P'. 

Through  P,   or  P,   draw  a    parallel   to    GB,  and  it  will 
be  the  tangent  required. 

1.  We  see,  from  this  construction,  that  if  two  tangents 
be  drawn  to  the  ellipse  through  the  two  extremities  of 
the   same   diameter,   they  will  be  parallel  to  each  other. 


To  draw  a  tangent  through  a  point  without  the  curve. 


J7  ,-' 


26.  Let  M  be  the  given 
point.  With  either  focus, 
as  jF",  as  a  centre,  and  a 
radius  equal  to  the  trans- 
verse axis,  describe  the  arc 
AW£r',  Then,  with  M 
as  a  centre,  and  a  radius 
equal  to  MJ\  the  distance 
to  the  other  focus,  describe 
the  arc  FKHK\  intersect- 
ing the  former  in  K  and 
K\    Through  K,   draw  KF' \  and  through  P,  where  it 


BOOK   ni.  THE      ELLIPSE.  103 

intersects  the  ellipse,  draw  the  straight  line  MPT^  and  it 
will  be  tangent  to  the   ellipse,   at   P. 

For,  since  P  is  a  point  of  the  ellipse,  F' P  +  PF  is 
equal  to  the  transverse  axis.  But  F' P  -f  PK  is  equal  to 
the   transverse  axis,  by  construction.     Hence,   PF  —  PK. 

Further,  since  the  arc  FK  is  described  from  the  centre 
M,  MF  =  MK;  hence,  the  line  MP  has  two  of  its 
points  each  equally  distant  from  the  pomts  F  and  PT;  it 
is,  therefore,  perj^endicular  to  FF* ;  and  since  the  triangle 
FPF  is  isosceles,  M  T  will  bisect  the  vertical  angle  P. 
The  opposite  angle  F'PM,  being  equal  to  TPF,  is  equal 
to  FPT;   hence,  MT  is  tangent  to  the   ellipse    (Art.  19). 

1.  The  two  arcs  KHK .,  KNK'.,  will  intersect  each  other 
in  two  points,  K  and  K' .  There  will,  therefore,  be  two 
lines,  KF\  K' F\  drawn  to  the  focus  F ;  hence,  there  will 
be  two  points  of  contact,  P,  P\  and,  consequently,  two 
tangent  lines,  il/P,  MP\ 

CONJUGATE     DIAMETERS. 

St.  Two  diameters  of  an  ellipse  are  said  to  be  conju- 
gate to  each  other,  when  either  of  them  is  parallel  to  the 
two  tangents  drawn  through  the  vertices  of  the  other. 

Since  two  supplementary  chords  may  be  drawn,  respect- 
ively parallel  to  any  diameter  and  the  tangent  through  its 
vertex  (Art.  2i),  it  follows,  that  two  supplementary  chords 
lufiy  always  be  drawn  respectively  parallel  to  any  two  con- 
jugate diameters. 

If,  therefore,  we  designate  by  a  and  a',  the  tangents 
of  the  angles  which  two  conjugate  diameters  make,  re- 
Bpectively,    with    the    transverse    axis,    these    tangents    will 

*  Legendre  Bk  L     Prop.  16.    Cor. 


104 


A>-ALTTICAL      GEOifETEY.  f BOOK   UI. 


fulfill  the  condition  of  sup- 
plementary chords,  and  sat- 
isfy the  equation, 


act'  = 


A} 


Let  us  designate  the  cor- 
responding angles  by  a  and  a.'    "We  shall  then  have, 


sm  a 
cos  a 


and     a'  = 


sm  a' 
cos  a' 


Substituting  these  values  in  the  last  equation,  and  reducing, 

we  obtain, 

A"^  sin  a  sin  a'  4-  -Z>-  cos  o  cos  a.'  =   0, 

and   dividing  both  members  by   cos  a  cos  a',  we  have, 

A^  tan  a  tan  a'  +  B-  =  0, 

an  equation,  which  expresses  the  relation  between  the  angles 
which  two  conjugate  diameters  form  with  the  transverse 
axis.  It  is  called,  the  equatioji  of  condition  of  conjugate 
diameters. 

In  the  equation  of  condition,  a  and  a'  are  undetei-rained. 
Hence,  any  value  may  be  assigned  to  either  of  them ;  and 
when  assicjned,  the  value  of  the  other  can  be  determined 
from  the    equation    of  condition. 

If  a  =  0,  we  shall  have,  sin  a  =  0,  and  cos  a  =  1. 
Hence,  JP  cos  a'  =  0,  and,  consequently,  cos  a'  =  0 ;  or, 
a'  =  90°.  Therefore,  when  one  of  the  conjugate  diame- 
ters coincides  with  the  transverse  axis,  the  other  will  coin- 
cide with  the  conjugate  axis.  Tlie  axes,  therefore,  fulfill 
the  condition  of  conjugate  diameters,  as  they  should  do, 
since  each  is  parallel  to  the  tangents  drawn  through  the 
vertices  of  the   other. 


i 


THE      ELLIPSE. 


105 


BOOK  IIl] 

Ellipse  referred  to  its  centre   and  conjugate  diameters. 
28.     The   equation   of  the  ellipse,  referred   to   its   centre 

Ahf  +  i?2ic2  =  A^B^ 

B 


and   axes,  is. 


Let  B'B    and    DD\   be    two 

conjugate   diameters. 

It  is  required  to  refer  the  el- 
lipse to  these  as  a  system  of 
oblique  axes,  and  to  find  its  equa- 
tion. 

The  formulas  for  passing  from  a  system  of  rectangular 
to  a  system  of  oblique  co-ordinates,  the  origin  remaining 
the   same    (Bk.  I.,  Art.  29),   are, 

X  z=  x'  cos  a  -f  y'  cos  a',  y  =  ic'  sin  a  +  y'  sin  a'. 

Squaring  these  values  of  x  and  y,  and  substituting  in 
the  equation  of  the  ellipse,  we  obtain  the  equation  of  the 
curve,  referred  to   conjugate   diameters ;  viz., 


(  +2(^2  gin  a  sin  a'-f  B'  cos  a  cos  a')x'y'  ) 


A^B--. 


But  the  equation  of  condition,   that  the  new  co-ordinate 
axes  shall  be   conjugate   diameters   (Art.  27),   is, 

A^  sin  a  sin  a'  4-  J>2  cos  a  cos  a'  =  0  ; 

tcnce,  the  equation  reduces  to, 

(.12 sin2 a'-f  J52 cos2 a')y'2_(_  (^^2 ^^^2 a -{■  B"^ cos2 a)x^=  A^B^. 

To  find  the   semi-diameter    (7-Z>,   make  y'  =  0 ;  then, 

A^B' 


A"^  sin2  a  -f  J52  cos2  a 
5* 


=    CB^  =   CB'^  =  A' 


106  ANALYTICAL      GEOMETllY.  [cOOKIH. 

If  we  make    x'  —  0,    we   shall  have, 

^  A-  sin^  a'  -f  M^  cos2  a' 

The  denominators,  in  the  two  last  equations,  are  the 
coefficients  of  a'^,  y'^^  {^  i}^q  equation  of  the  curve,  referred 
to  oblique  axes.  Finding  their  values,  and  substituting  them 
in  that  equation,   Ave  have, 

y'2  ^.'2 

;|7i  +  -^   =  1  ;    hence, 

or,   omitting  the  accents  of  x  and  y,  since  they  are  general 

variables, 

^'2?/2  +  J5'2a.2  _   A'^j^'^, 


which  is  the  equation  of  the   ellipse,  referred  to  its  centre 
and   conjugate   diameters. 

This  equation,  being  of  the  same  form  as  the  equation 
of  the  ellipse,  referred  to  its  centre  and  axes,  it  follows, 
that  every  value  of  x  will  give  two  equal  values  of  y, 
with  contraiy  signs;  and  every  value  of  y,  two  equal 
values  of  x,  with  contraiy  signs;  hence,  the  ellipse  is 
symmetrical  with  respect  to  either  of  its  conjugate  dia- 
meters ;  that  is, 

Either  diameter  bisects  all  chords  drawn  parallel  to  the 
other  and  terminated  hy   the  curve. 

Relation  of  ordinates  to  each  other. 

29.  The  equation  of  the  ellipse,  referred  to  its  conjugate 
diameters,   is, 


THE      ELLIPSE. 


107 


BOOK  III.] 

A'Y  +  ^'^^^  =  A'^B'\ 

If  we  designate  any  two  or- 
dinates  to  the  diameter  AB^  by 
y\  y'\  and  the  corresponding 
abscissas    by    x\    x'\    we    shall 

have, 

y^  _  {A'  +  x'){A'  ~x') 

y"2    -    {A'  -{-  X"){A'  -  X")' 

7/2    :    2/''2    ..    {A' +  x'){A' -  X')    :    {A' -\- x"){A' -  x"). 

If  the  ordinates  be  drawn  to  the  conjugate  diameter, 
it  may  be  readily  shown,   that, 

x"'    :    x"^    ::    {B' ^  y'){B' -  y')    :    (B' +  y"){B' -^  y") 

Hence,  tlie  squares  of  the  ordinates  to  either  of  two 
conjugate  diameters^  are  to  each  other  as  the  rectangles  of 
the  segments  into  which  they   divide  the  diameter. 


Parameter. 

30.  The  Parameter  of  any  diameter  is  a  third  propor- 
tional to  the  diameter  and  its  conjugate.  Thus,  if  P  desig- 
nate the  parameter  of  the  diameter  2A\  we  shall  have, 

2A'    :    2B'    '.:    2B'    :    P, 

2B'^ 

A'  ' 


or. 


P  = 


Relations  between  the  axes  and  conjugate  diameters. 

31.     The   equation    of   the   ellipse,    referred    to    its    con- 
jugate  diameters,  which   are   oblique   axes,   is 


108  ANALYTICAL       GEOMETKY.  [jJOOK   lU. 

It  is  required  to  refer  the  ellipse  to  its  transverse  and 
conjugate  axes,  which  is  a  rectangular  system. 

The  formulas  for  passing  from  oblique  to  rectangular  axes, 
the  origin  remaining  the  same  (Bk.  I.,  Art.  30),  are, 

a  sin  a'  —  2/  cos  a'  y  cos  a  —  x  sin  a 

X  =■ 2/'= 

sin  (a'  —  a)  sin  (a'  —  a) 

Substituting  these  values  of  y\  x\  we  have, 

(^'2  cos^a  +^'2  cos2a')  2/2  4.  (^'2  ^\^2a,  +  ^'2  sin2a')a;2 
—  2{A'^  sin  a  cos  a  +  B'^  sin  a'  cos  a')  xy  =  A'^  JB'^  sin2  (a'—  a), 

which  is   the  equation   of  the  ellipse,  referred   to   its   centre 
and  axes,  and,  hence,  must  be  of  the  form, 

Jl^^  -\-B^  x^  =AB^;    ....     (1.) 

consequently,         A'^  sin  a  cos  a  +  J?2  gJQ  q^'  qq^  ^  _  q^ 

and  we  have, 

(.tt'2  cos2a  +  B"^  cos2a')2/2  +  ^A'"^  sin2a  +  B'^  sin2a')a;2  = 
^'2i?'2   fein2  (a'  -  a) (2.) 

If  we  multiply  together,  the  values  of  A"^  and  -S'2  (Art. 

2§),  we  have, 

A'^B"^   equal  to 

A'^B^. 

^<  sin'a' ein'a   +   J?  B\^\x:?aQ,Q'S?a'  +  cos'asin'a')   +   ^*cos'aCOsV, 

which  may  be  put  under  the  form, 

A^BK 

{A?  sin  a'  sin  a  +  B"^   cos  a!  cos  a)  2  -f  ^2^2  31^2   (ct'—  a) 

after  adding  and  subtracting,  in  the  denominator, 
2^^-52~(sifr»^^€0S»rt^^ii-  <;os^^sin^a'). 


BOOK   III.]  THE      ELLIPSE.  109 

But  the  first  terra  ol  the  denominator  is  0  (Art.  27) ; 
hence, 

^/2  £'2   ^  £_J? =—^^ . 

^2J52  sin2  (oc'-  a)       sin2  (a'-  a) 
Whence,  A^^^  =  A'^B'^  sin2  (a'  -  a). 

The  second  members  of  equations  (1)  and  (2)  are, 
therefore,  equal,  and  by  making  x  and  y  each  equal  to  0, 
in  succession,  in  equation  (2),  we  prove  that  the  co-efiicients 
of  y^  and  a^,  in  equations  (1)  and  (2),  are  also  equal,  each 
to  each;  hence, 

^''cos^a  +  ^'^cosV^^^    .     .     .     (1.) 

A'^  sin'^a  +  B"  sinV'z=  B'    .     .     .     (2.) 

A"  B"  sin*^  (a'-  a)  =  A'B\ 

or,  A'B'   sin    (a'  -  a)  =  AB      .     .     (3.) 

The  equation   of  condition,  of  conjugate  diameters   (Art. 

27),  is, 

A'  tan  a  tan  a'  +  B^  =  0. 

If  we  add  Equations  (1)  and  (2),  member  to  member,  and 
recollect,  that  cos'^a  +  sin'a  =  1,  we  have, 

A"-\-B''  =  A'+B\ 

Tabulating  the  last  three  equations,  and  changing  their 
order,  for  convenience  of  interpretation,  we  have, 

A'  tan  a  tan  a'+  jB^  =  0 (1.) 

A'B'  sin  (a'-  a)  =  AB (2.) 

A''  +  B"  =  A'  +  B'      ...     (3.) 

Tliese  three  equations  express  the  relations  that  exist  be- 
tween a  and  a',  the  semi-axes,  A  and  B,  and  any  two 
semi-conjugate  diameters.  A'  and  B'. 


110 


ANALYTICAL      GEOMETRY.         [bOOK    111. 


Interpretation  of  A'^  tan  a  tan  a'+W  =  0. 

1.  If  we  know  the   angle  which  the   conjugate  diameters 
make  with  each   other,  it  will  be   equivalent  to  knowing   a 
or  a'.     For,  denote  the  known  angle  by  /3; 
then  a'— a  =  i3; 

or  a'  =  (3  -\-  a-        hence, 

,  _         .  tan  /3  +  tan  a 

tan  a'  =  tan  (/3  +  a)  = ^— 

^  ^       1  —  tan  p  tan  a 

Substituting  this  value  of  tan  a',  in  Equation  (1),  we  have, 

A'  tan'^a  +  (A'  -  B')  tan  a  tan  i3  +  ^*  =  0 ; 

from  which  we  can  find  tan  a,  and,  consequently,  a,  in  terms 
of  the  axes  and  the  known  quantity,  tan  ^S. 


Interpretation  of  A'B'  sin  (a'—  a)  =  AB, 

2.  Let  us  suppose  the 
ellipse,  whose  centre  is  (7, 
to  be  circumscribed  by  a 
rectangle,  formed  by  draw- 
ing tangents  at  the  verti- 
ces of  the  axes,  and  also 
by  a  parallelogram,  formed 
by  drawing  tangents  at  the 
vertices  of  the  conjugate 
diameters.  Denote  the  semi-conjugates,  CP  and  CiV,  by 
A'  and  B'. 

From   Jf,   Draw  MK  perpendicular  to    CN'.    The   angle 
JSTCP   is   designated   by   a'  —  a,  and    since   MN^G  is    the 


BOOK   III.]  THE      ELLIPSE.  Ill 

supplement  of  NCP^  its  sine  will  be   equal   to   sm  (a'—  «). 
Further,  NM  =z  CF=A'.     Therefore,* 

3fK=  A'  sin  (a'-  a). 
Hence,  AB'  sin  (a'-  a)  =  GP3IN.\ 

The  second  member  of  Equation  (2),  ^  x  ^,  is  equiva- 
lent to  the  rectangle  GJBHD.  But  the  parallelogram 
CPMN  is  one-fourth  the  parallelogram  3/Z,  and  the  rect- 
angle CBIID  is  one-fourth  the  rectangle  IIF\  hence, 
Equation  (2)  expresses  the  following  property: 

The  rectangle  which  is  formed  hy  drawing  tangents 
through  the  vertices  of  the  axes^  is  equal  to  the  paral- 
lelogram which  is  formed  hy  drawing  tangents  through  the 
vertices  of  two  conjugate  diameters. 


Interpretation  of  J.''  4-  B''  —  A  ■\-  B\ 
3.  If  we  multiply  both  members  by  4,  we  have, 
4^"  +  ^B"  =  W  +  ^B\ 
which  expresses  the  following  property: 

The  sum  of  the  squares  described  on  the  axes  of  an  el- 
lipse^ is  equal  to  the  sum  of  the  squares  described  on 
any  two  conjugate  diameters. 

"  The  area  of  the  ellipse  is  found  in  the  Calculus,  p.  75. 

*  Legendre,  Trig.  Art.  37»v'/j^%  f  Mens.  Art.  95. 


BOOK    lY. 


d — 

c 

jB 

OF      THE      PARABOLA. 

1.  The  Pakabola  is  a  plane  curve,  such  that  any  point 
of  it  is  equally  distant  from  a  fixed  point  and  a  given 
straight  line. 

The  fixed  point  is  called  the  fo- 
cus of  the  parabola,  and  the  given 
straight  line,  the  directrix. 

Thus,  if  i^  be  a  fixed  point,  and 
£^D  a  given  line,  and  the  point 
P  be  so  moved,  that  Pi^  shall  be 
constantly  equal  to  P(7,  the  point 
P  T\-ill  describe  a  parabola,  of  which 
7^  is  the  focus,  and  DU  the  di- 
rectrix. 

1.  This  property  of  the  parabola  afibrds  an  easy  metliod 
of  describing  it  mechanically. 

Let  PX  be  a  given  line,  and  JLCD 
a  triangular  ruler,  right-angled  at 
C.  Take  a  thread,  the  length  of 
which  is  equal  to  the  side  (7P,  and 
attach  one  extremity  at  P,  and  the 
other  at  any  point,  as  P.  Place  a 
pencil  against  the  thread  and  the 
ruler,  making  tense  the  parts  of  the 
thread  PP.  PP.     Then,  if  the  side 


BOOK  TV,] 


THE      PARABOLA, 


113 


(7X  of  the  ruler,  be  moved  along  the  line  J5X,  the  pencil 
will  describe  a  parabola,  of  which  JF  is  the  focus,  and  J5i 
the  directrix ;  for,  the  distance  JPJ^  will  be  equal  to  P  C, 
for  every  position  of  the  ruler. 


Equation  of  the  Parabola. 

2.    Let  JF^  be  tbe  focus,  and  DC  . 

the  directrix.  Denote  the  distance 
i^,  from  the  focus  to  the  direc- 
trix, by  p,  and  let  the  point  .1, 
equally  distant  from  2^  and  J^  be 
assumed  as  the  origin  of  a  system 
of  rectangular  co-ordinates,  of  which 

AX,  AY,   are  the   axes.    The  dis- 

♦) 
tance  AJ^,  will  be  denoted  by   -• 

z 

Let  P  be  any  point  of  the  curve,  and  denote  its  co-ordi- 
nates by  X  and  y. 

Then,  the  distance  between  any  two  points  (Bk.  1.  Art.. 
1»),  is,  

v^"  -  ^r  +  (y"  -  ^y 

Substituting  for  x'\  y",  the  co-ordinates  of  the  point  P, 
which  are  x  and   y,  and  for  x',  y',   the   co-ordinates  of  F^ 

which  are,  a/  =  "^j   and  y'  —  0,  we  have. 


FP  =  yV+(7-|) 

But,  by  the  definition  of  the  curve, 
FP  =  PC  =  PA  -{-  AB 


1  +  ^ 


'(I.) 


I 


114  ANALYTICAL      GEOMETRY.  [bOOKIV. 

Hence,  sj y-^  _f.  ^a;  -  1^  =  |  +  aj , 

or,  2/2  +  a.2  _  j^  ^  ^  _  ^  ^  ^  _j_  a.2. 

hence,  y^  =  Ipx, 

which  is  the  equation  of  the  parabola,  referred  to  its  vertex 
and  the  rectangular  axes,  AX^  and  -4  Yi 

Interpretation  of  the  equation. 

3.  The  axis  of  abscissas,  AX^  is  called  the  axis  of  the 
parabola,  and  the  origin  -4,  is  called  the  vertex  of  the  axis, 
or  principal  vertex. 

1.  The  equation  of  the  parabola  gives, 

y  =   ±:  ^2px. 

from  which  we  see,  that  for  every  positive  value  of  aj, 
there  will  be  two  equal  values  of  y,  with  contrary  signs. 
Hence,  the  parabola  is  symmetrical  with  respect  to  its  aa^is. 

2.  "We  see,  further,  that  y  will  increase  with  a;,  and  will 
have  real  values  so  long  as  x  is  positive.  Hence,  the  curve 
extends  indefinitely^  in  the  direction  of  x  positive. 

If  we  make  a;  =  0,   we  have, 

2/  =   ±0, 

which  shows,  that  the  axis  of  I^  is  tangent  to  the  curve,  at 
the  origin. 

If  we  make  x  negative,  we  shall  have. 


y  =  ±  -/- 2/)aj; 


BOOK   IV.]  THE      PAEABOLA.  115 

or,  y  imaginary ;  which  shows,  that  the  curve  does  not  pass 
the  axis  of  Y,  and  extend  on  the  side  of  x  negative, 

3.  By  a  course  of  reasoning  similar  to  that  in  Bk.  IIL, 
(Art  4 — 5,)  we  have  the  conditions  for  determining  the  posi- 
tion of  a  point,  with  respect  to  the  cm*ve.    They  are, 

For  a  point  without  the  curve,         y^  _  2px  >  0. 
For 'a  point  in  the  curve,  y^  —  2px  =  0. 

For  a  point  within  the  curve,  y^  —  2px  <  0. 

"^^JL  Parameter. 

4,  The  Parameter  of  the  axis,  is  the  double  ordinate 
through  the  focus. 

1.  If,  in  the  equation  of  the  parabola, 

y"^  =  2px, 

we  make,    x  =  ^,    the  corresponding  value  of  y,  will  be 

the  ordinate   through    the  focus.      Under  this  supposition, 
we  have, 

y2  =  2p  X  I  =  jp2 ;    or,    y  =  p. 

Hence,  2p  =  the  parameter. 

2.  In  the  ellipse,  the  parameter  of  the  transverse  axis 
is  ar  third  proportional  to  the  axes  (Bk.  m..  Art.  11);  in 
the  parabola,  it  is  a  third  proportional  to  any  abscissa 
and  the  corresponding  ordinate.    For,  from  the  equation, 

2/2  =  2px, 
we  have, 

X    :    y     ::     y    :     2p. 

3.  If  the  parameter  and  axis  of  the  parabola  are  known, 


116 


ANALYTICAL      GEOMETKY.  [bOOK   IV. 


we  have  a   simple   construction   for   detennining    points  of 
the   curve. 

Let  ^JT,  AYj  be  the  co-or- 
dinate axes.  The  equation  of 
the  curve  is, 

From  the  origin  A,  lay  off 
a  distance  AB,  on  the  nega- 
tive side  of  abscissas,  equal  to 
2p.  Then,  from  A,  lay  off  any  distance,  as  AP,  and  draw 
PM  perpendicular  to  AJi.  On  -SP,  as  a  diameter,  de- 
scribe a  semi-circumference,  and  through  <?,  where  it  inter- 
sects the  axis  AY,  draw  QJT  parallel  to  A^ :  The  point 
JIT,  where  it  intersects  P3f,  will  be  a  point  of  the  curve. 
For,   from  the   equation   of  the  circle, 

AQ''  =  BA .  AP, 
hence,  y"^  =  Ipx, 

for   any  point,    M  or    M', 


^ 


Relation  of  the  ordinates  and  abscissas. 


5.  Denote  any  two  ordinates  of  the  curve,  by  y'  and 
y'\  and  the  corresponding  abscissas,  by  x'  and  x" .  We 
shall  then  have,   from  the   equation   of  the   curve, 

2/'2  =  2px\       and      y""-  —  2px" ; 

hence,  y''^    :    y"^    :  :    x'    :    x'\ 

by   omitting   the   common  factor   2/>;   that  is. 

The  squares  of  the  ordinates  are  to  each  other  as  their 
corresponding  abscissas. 


BOOK  IV.] 


THE      PAKABOLA. 


117 


Polar  Equation. 

6.     Let  US    resume    the    consideration    of  Equation   (1) 
(Art.  2),   which  is, 

FP  =  r  =^_^x (1.) 


T    =l  +  X, 


and  in  which  the  origin  of  co-ordi- 
nates is  at  the  vertex  of  the  axis. 
The  formulas  for  transferring  the 
origin  to  the  focus,  whose   co-ordi- 


nates   are,   a 


P 


,    and    h 


(Bk.  L,  Art.  2§),  are, 


X  =^  +  x\     and     y 


y'^ 


Substituting  this  value  of  ic,   in  Equation  (1),  we  Lave, 

p  ■{-  x'    .     .    .     (2.) 


-1+1  +  -' 


If  we   denote  the  variable  angle  which  the   ladius-vector 
makes  with  the  axis,  by  v,  we  have, 

x'  =  r  cos  V ;    hence,    r  =  p  ■\-  r  coo  c ; 


whence. 


P 


1—  cosv 


(3.) 


which  is  the  polar  equation  of  the  parabola,  when  the  pole 
is  at  the  focus. 

1.  We  see,  from  Equation  (2),  that  the  radius-vector 
is  expressed,  rationally,  in  terms  of  the  abscissa  of  the 
point  in  which  it  intersects  the  curve.  This  property  is 
peculiar  to  the  focus. 


118  ANALYTICAL      GEOMETEY.  [bOOK  IV. 

Interpretation  of  the  polar  equation. 
7,    In  the  polar   equation, 

P 


1  —  cosv' 


as  well  as  in  the  corresponding  equation  of  the  ellipse, 
which  is  expressed  under  a  similar  form  (Bk.  IH.,  Art.  O-l), 
the  values  of  the  radius-vector  begin  at  the  remote  vertex, 
that  is,  in  the  case  of  the  parabola,  at  an  infinite  distance 
from  the  focus. 

If  we  make    v  =  0,    we  have, 

r  -  -  =  CO. 

If  we  make    v  =  90°,    we  have, 

r  =  p. 

that  is,   half  the  parameter. 

If  we  make    v  =  180°,    we  have, 

1.    If  it  is  desirable  that   the  values   of  r  should  begin 
at  the  nearest  vertex,  make    v  =  180°  —  v\    and  we  shall 

have, 

cosv  =  —  cosu'. 

Substituting,    —  cos  v'    for    cos  v,    the  equation  becomes, 

P 

1  +  cos  v' ' 

in  which   equation,   the  values  of   r  begin  at  the  nearest 
vertex,  and  v'  increase  from   0   to   360°. 


BOOK  IV.] 


THE      PARABOLA. 


119 


Tangent  line  to  the  parabola. 

8.  Let  us  designate  the  co-ordi- 
nates of  any  point  of  the  curve,  as 
P,  by  jc",  y"  \  the  equation  of  a 
straight  line  passing  through  this 
point    will    be, 

y-y"  =  a{x-x")    .  .  (1.) 

It  is  required  to  determine  a,  when 
the  right  line  is  tangent  to  the  para- 
bola.    The  equation  of  the  parabola  is, 

y^  =  <lpx', 

and,  since  the  point  of  tangency  is  on  the  curve,  we  also 

have, 

y"2  _  2px", 

Subtracting  the  last  equation  from  the  preceding,  we  ol>- 
tain, 

(y  +  y"){y-y")  =  2p{x-x")   .  .   (2.) 

Combining  Equations  ( 1 )  and  ( 2 ),  we  have, 

{y  +  y")  a{x-  X")  =  2p{x  -  W') ; 

or,  transposing   and  factoring, 

{x-x'')[a{y  +  y^')^2p]  =  0; 

an  equation  which  may  be  satisfied  by  making, 

X  —  x"  =  0,         or,         a(y  +  y")  —  2p  =  0. 

In   the   first    equation,    x   is    the    abscissa    of   P;    in    the 
second,  y  is  the  ordinate  of  7*';\vlKn  P'  becomes  consecutive 


120 


ANALYTICAL      GEOMETRY.  [bOOK    IV. 


with  P,  we  have, 


n-    P 

a  =  —rr 

y 


Substituting  this  vahie,  in  the  equation  of  the  line  passing 
through  jP,  we  ha^e, 

y-y"=  ~r,{^ -''")■' 

and,  by  reducing,  and  observmg  that  y""^  =  2pa5", 

yy"  =  p{^  +  «")> 

which  is  the  equation  of  the  tangent. 


Sub-tangent. 
9.     If,  in  the  equation  of  the  tan- 

yy"  =  p{^  4-  X"), 

we  make   y  =  0,   we  shall  have, 

0  =  2^^  +  »") ; 

but  since  the  factor  p,  is  a  constant 


gent. 


quantity, 


T     A 


X  -\-  x' 


0 ;        or,        jc' 


rr  


that  is,  AD^  the  abscissa  of  the  point  of  tangency,  is  equal 
to  —  AT  \  or,  the  sub-tangent  7!Z>,  is  bisected  at  the 
vertex  A. 

The  analytical  condition,  expressed  by 

x  +  x"  =  0,        or,        AT  +  AD  =  0, 

indicates,  that  the  quantities  are  numerically  equal  with 
contrary  signs  ;  hence,  they  are  estimated  on  different 
sides   of   the   oriirin. 


BOOK  IT.] 


THE      PA  R  ABOL  A. 
Normal  and  Sub-normaL 


121 


10.  Let  x'\  y"^  be  the  co-ordmates 
of  the  point  of  tangency.  Then,  the 
equation  of  the  normal  will  be  of  the 
form, 

y-y"  =  a\x  -  x"\ 

and    since  it  is  perpendicular  to  the 
tangent, 

aa'  +  1   —  0. 
But  we  have  already  found  (Art.  8), 


V 

y' 

therefore,  we  have, 


hence. 


r 
p 


y 


y-y"=  -  ^('^  -  «"), 

which  is  the  equation  of  the  normal. 
1.    If,  in  the  equation  of  the  normal, 

y-y"=  -  ^'(^  -  «"),      * 

we  make  y  =  0,   and  then  find  the  value  of  a;  —  a",  we 
shall  have, 

X  —  x"  =  p. 

But,  X  is  equal  to  the  distance  AN"^   and  x"  to  the  dis- 
tance AR ;  hence,  x  —  x"  =  RN'  =  p ;  that  is,   the  sub- 
normal is  constant,  and  equal  to  half  the  parameter. 
6 


1-J2 


ANALYTICAL      GEOMETRY. 


[book  rv. 


Ferpendiciilar  from  the  focus  to  the  tangenL 
11.     The  equation  of  a  line  passing  through  the  focus, 
whose  co-ordinates  are,   a'  =  ^ ,   and  y'  =  0,  is, 


y  =  4-f) 


The   condition,   that  this    line   shall    be  perpendicular  to 
the  tangent,  gives, 

aa'  -^  1   =  0; 


hence, 


P 


y 


the    equation    of    the    perpendicular 
HF^  therefore,   becomes. 


»  =  -*-!) 


P 

Combining  this  with  the   equation 
of  the  tangent   TP,   which  is, 

and  substituting  for  y""^^  its  value   2/>aj",  and  reducing,  we 

find, 

x{1x"  +  ^)  =  0  ; 

an  equation  which  can  only  be  satisfied  when  a;  =  0 ;  hence, 
the  point  H^  at  which  the  perpendicular  meets  the  tangent^ 
is  071  the  axis  of  Y. 

12,     Through  P,  the  point  of  contact,  draw  PD  perpen- 
dicular to  the  axis ;  then  (Art.  9), 


TA  =  AD ;        and  hence,        TIT  =  HP, 


THE      PARABOLA. 


123 


HOOK  IV.] 

Therefore,  the  two  right-angled  triangles  TFII  and  IIFP^ 
have  the  two  sides  about  the  right  angle  equal ;  consequent- 
ly, the  triangles  are  equal,  and  the  angle  FTP  is  equal  to 
TPF\  that  is. 

The  tangent  to  the  parabola  at  any  point  of  the  curve, 
makes  equal  angles  with  the  axis  and  with  the  line  drawn 
from  the  point  of  tangency  to  the  focus. 

13.  In  the  right-angled  triangle  TFH,  in  which  AS 
is  perpendicular  to  TF^  we  have,* 


FH    =  FTx  FA', 


or. 


FH 


FP  X  FA, 


But,  FA  is  equal  to  —p\  hence,  it  is  constant  for  every 

position  of  the  point  of  contact ;   thbrefore,   FlP  varies  as 
the  distance  FP  \  that  is, 

The  square  of  the  perpendlcidar  drawn  from  the  focus 
to  the  tangent,  varies,  as  the  distance  from  the  focus  to 
the  point  of  contact. 


CONSTRUCTION      OF      TANGENT      LINES, 
Tangent  line  at  a  given  point  of  the  curve. 


14.— 1. — First  Method.  Let  P  be 
the  given  point.  Draw  PR  perpen- 
dicular to  the  axis.  Then,  from  the 
vertex  A,  lay  oS  AT  equal  to  AE, 
and  join  T  and  P ;  TP  will  be  tan- 
gent to  the  curve   (Art.  9). 


»  Legendre,  Bk.  IV.     Prop.  23. 


124 


ANALYTICAL      GEOMETRY 


[book  IV. 


2. — Second  Method.  Draw  the  or- 
dinate Pi?  to  the  axis,  and  from  the 
foot  P,  lay  off  a  distance  Hy  =  p, 
and  join  P  and  JST.  Then,  draw  TF 
perpendicular  to  PiV  at  P,  and  it 
will  be  the  tangent  required  (Art. 
10). 

3. — Third  Method.  Join  P  and  the  focus  P'  (next  figure). 
Then  lay  off  from  JF]  on  the  axis,  a  distance  FT,  equal  to 
PP,  and  join  P  and  T;  FT  will  be  the  required  tangent. 
(Art.  12.) 


Tangent  parallel  to  a  given  line. 

15.     Let  P(7  be  a  given  line,  to  which  a  tangent  is  to 
be  dra^-n  parallel.     At  the  focus  F, 
lay  off  an  angle  XFF,  equal  to  twice 
the  angle  which  the  given  line  makes 
with  the  axis  of  X. 

Through  P,  the  point  at  which 
FF  intersects  the  curve,  draw  FT 
parallel  to  P(7,  and  it  will  be  the 
tangent  required. 

For,  the  outward  angle  FFX  is 
equal  to  the  sum  of  the  angles  T  and  PPP*  But  FTF 
being  equal  to  the  angle  which  BC  makes  with  the  axis 
of  X,  is  equal  to  one-half  of  FFX  \  hence,  the  angle  FTF 
is  equal  to  half  of  FFX\  therefore,  the  triangle  FTF  is 
isosceles,  and,  consequently,  FT  is  tangent  to  the  curve  at 
P  (Art.  12). 

*Legendre,  Bk.  I.    Prop.  25.     Cor.  6. 


BOOK   ^V^]  THE      PARABOLA. 

""'^ent  through  a  given  point  without  the  curvew 


125 


Let  6^  be  a  given  point, 
through  which  a  tangent  is  to  be 
drawn. 

With  (r,  as  a  centre,  and  a  radius 
equal  to  GF^  the  distance  to  the 
focus,  describe  the  arc  of  a  circle 
intersecting  the  directrix  at  C  and 
C.  Through  C  and  C\  draw  two 
lines  parallel  to  the  axis  BX^  inter- 
secting the  parabola  in  F  and  F .  Through  G^  draw 
GF  and  GF^  and  they  will  be  tangents  to  the  parabola, 
at  F  and  F' . 

For,  join  F  and  the  focus  F.  Then,  since  P  is  a  point 
of  the  parabola,  FF—  FC\  and,  by  construction,  GF=  GC\ 
hence,  the  line  GF  has  two  points,  G  and  P,  each  equally 
distant  from  C  and  P;  it  is,  therefore,  perpendicular  to 
(7P*  Since  the  triangle  OFF  is  isosceles,  FG  bisects 
the  angle  (7PP;  therefore,  TFF  =  FTF ',  hence,  TF  is 
tangent  to  the   curve   (Art.  12). 

It  may  be  proved  that    GF'  is  tangent  at  F', 


PARABOLA   REFERRED   TO   OBLIQUE   AXES. 

,17.  We  have  thus  far  deduced  the  properties  of  the 
parabola,  from  its  equation,  obtained  by  referring  the  cui-ve 
to  a  system  of  rectangular  co-ordinates,  having  their  origin 
at  the  vertex.  We  now  propose  to  develop  some  of  the 
properties  of  the  curve,  by  referring  it  to  a  system  of 
oblique   co-ordinates. 

*  Legendre,  Bk.  I.    Prop.  16.    Cor. 


126  ANALYTICAL      GEOMETEY.  [bOOK  IV. 

Sqoation  when  referred  to  oblique  axes. 

18.  The  formulas  for  passing  from  rectangular  to  ob- 
lique co-ordinates,  when  the  origin  is  changed  (Bk.  I., 
Art.  29 — 1),  are, 

X  =  a  +  ic'  cos  a  +  y'  cos  a',        y  =  J  +  a'  sin  a  +  y'  sin  a' 

Substituting  these  values  of  x  and  y,  in  the  equation, 

y2  =  2px, 
it  becomes, 

(1.) 

y'2  sin2a'+  2x'y'  sin  a  sina'+  x'^  sin^a  +  b^—  2ap  )    __ 
+  2(6sina'— ^cosa')y'-f  2(5  sin  a  —  j9C0sa)aj'  ) 

In  this  equation,  there  are  four  arbitrary  constants,  a, 
b,  oL  and  a',  to  which  we  can  assign  values  at  pleasure. 
By  giving  a  fixed  value  to  either,  we  introduce  one  con- 
dition into  the  coefficients  of  Equation  ( 1 ),  and  by  assigning 
values  to   all,   we  introduce  four  conditions. 

Let  the  values  assigned  to  these  arbitrary  constants  be 

such,    that    Equation    (1)    shall    contain    only   the    second 

power  of  y,  and  the  fii'st  power   of  x;  that  is,  reduce  to 

the  form, 

2/2  —  2/XB. 

This  requires  that, 

b^  -  2ap  =  0 (1.) 

sin^a  =  0 (2.) 

an  a  sin  a'  =  0 (3.) 

ft  sin  a' —jo  cos  a'  =  0  .     .     .     .     .  (4.) 


BOOK  IV.] 


THE      PARABOLA, 


127 


Having  introduced  these  conditions,  the  equation  becomes, 


,'2    -       y    ^^ 

sin^a' 


Let  us  interpret  these  four  conditions,  separately. 

Interpret   the    equation,     52  —  2ap  =  0. 

1.  This  equation  of  condition 
is  of  the  same  form  as  the  equa- 
tion of  the  parabola,  referred  to 
the  primitive  axes.  Therefore,  the 
co-ordinates  of  the  new  origin  will 
satisfy  the  primitive  equation,  and 
hence,  the  new  origin  is  on  the 
curve  at  some  point  as  A\ 


y 


Interpret   the   equation,     sin^a  =  0. 

2.    In  this  equation   of  condition,  we  have, 

sin^a  =  0 ;        hence,         a  =  0, 

which  shows,  that  the  new  axis  of  abscissas,  -X"',  is  parallel 
to  the  primitive  axis  AX^, 


Interpret   the   equation,     sin  a  sin  a' 


0. 


3.  This  equation  of  condition,  is  satisfied  by  virtue  of 
the  sin  a  =  0 ;  hence,  it  is  nothing  more  than  the 
second. 


128  ANALYTICAL      GEOMETRY.  [bOOK    IT. 

Interpret    the    equation,     h  sina'  —  p  cosa'  =   0. 
4.  This   equation   of  condition,   gives, 

P 

tan  a'  =  Y ; 
o 

and  since  this  value  of  tana'  is  the  same  as  that  found 
in  (Art.  8),  for  the  tangent  of  the  angle  which  the  tan- 
gent makes  with  the  axis  of  JT,  we  conclude  that  the 
new  axis  y,  is  tangent  to  the  parabola  at  the  new 
origin,   A'. 

Interpret    the    equation,     y^  =     .   ^  /^'' 
19.    To  simplify  the  form,  put, 

we  shall  then  have,  by  omittmg  the  accents  of  the  variables, 

2/2  =  2p% 

for  the  equation  of  the  parabola,  referred  to  the  new  axes. 

2/? 
The    coefficient,    2p',    or    its    equal,     .  ^  ^,    is    called    the 

2)arameter,   of  the  new   diameter   A'^'. 

In  this  equation,  every  value  of  x  will  give  two  equal 
values  of  y,  with  contrary  signs ;  hence,  the  curve  is 
symmetrical  with  respect  to  the  axis  A'Jl'  ;  or,  this 
axis  bisects  all  chords  of  the  parahola  ichich  are  parallel 
to   the   tangent  A'  Y'. 

1.  Diameter^  as  a  general  term,  designates  any  straight 
line   which  bisects  a  system  of   chords  drawn    parallel  to 


BOOK   rv.]  THE      PARABOLA.  129 

the  tangent  at  the  vertex,  and  terminating  in  the  curve; 
and  the  curve  is  said  to  be  symmetrical  with  respect  to 
the  diameter,  whether  the  chords  are  oblique  or  perpen- 
dicular to  it.  In  this  sense,  therefore,  every  line  drawn 
parallel  to  the  axis  AX,  is  a  diameter  of  the  parabola; 
hence,  all  diameters  of  the  parabola  are  parallel  to  each 
other,  a  'property  lohh-h  shows  that  the  centre  of  the  curve 
is  at  an  infinite  distance  from  the  vertex. 

For  the  area  of  the  parabola,  see  Calculus,  page  72. 


EOOK    Y. 

OF      THE      HYPERBOLA 

1,  An  Hyperbola  is  a  plane  curve,  such  that  the  dif- 
ference, of  the  distances  from  any  point  of  it  to  two  fixed 
pomts,  is  equal  to  a  given  distance. 

The  fixed  points  are  called  the  foci. 

The  characteristic  property  of  the  hyberbola,  gives  rise 
to  the  following  constructions  of  the  curve. 


First — By  a  continuous  movemenL 

2.  Let  F'  and  F,  be  the  foci,  W 
and  FF^  the  distance  between 
them.  Take  a  ruler,  longer  than 
the  distance  F'F^  and  fasten  one 
of  its  extremities  at  the  focus  F' . 
At  the  other  extremity,  J?i  at- 
tach a  thread  of  such  a  length, 
that  the  length  of  the  ruler  shall 

exceed  the  length  of  the  thread  by  a  given  distance  AB. 
Attach  the  other  extremity  of  the  thread  at  the  focus  F, 

Press  a  pencil,  P,  against  the  ruler,  and  keep  the  thread 
constantly  tense,  while  the  ruler  is  turned  around  F'^  as 
a  centre.  The  point  of  the  pencil  will  describe  one  branch 
of  the  curve. 

For,  FF  +  PH  is  equal  to  the  length  of  the  thread,  to 


BOOK  v.] 


THE      HYPERBOLA. 


131 


which  if  we  add  AB,    we  shall    have  the  length  of  the 
ruler.     Hence, 


or, 


F'JP  +  JPJI  =  IT+  PjEr+  AB; 
F'P-  FP  =  AB; 


therefore,  P  is  a  point  of  the  hyperbola. 

If  the  extremity  of  the  ruler,  attached  at  the  focus  F^ 
be  removed,  and  attached  at  F,  the  second  branch  of  the 
hyperbola,  WAP^  may  be  described  in  a  similar  manner. 

3.  The  TRANSVERSE  AXIS  is  that  part  of  the  line  drawn 
through  the  foci,  lying  between  the  two  branches  of  the 
curve,  as  AB.  The  points  A  and  J5,  in  which  the  trans- 
verse axis  intersects  the  curves,  are  called   vertices, 

4.  The  CENTRE  of  the  hyperbola,  is  the  middle  point  (7, 
of  the  transverse  axis. 

5.  The  line  (7Z>,  drawn  through  the  centre,  perpendicu- 
lar to  the  transverse  axis,  and  equal  to  the  square  root  of 
OF  —  CB  ,  is  called  the  semi-conjugate  axis ;  and  DD\  is 
the  conjugate  axis. 


Second — Construct  the  curve  by  points. 


6.  Let  AB  be  a  given  line, 
ani  F'  and  F^  two  given  points. 
It  is  required  to  describe  an 
hyperbola,  of  which  AB  shall 
be  the  transverse  axis,  and  F' 
and  F^  the  foci. 


132 


ANALYTICAL       GEOMETBY. 


[book  V. 


From  the  focus  F,  lay  off  a 
distance  F'N^  equal  to  the  trans- 
verse axis,  and  take  any  other 
distance,  as  F'H^  greater  than 
F'B, 

With  F'  as  a  centre,  and  F'H 
as  a  radius,  describe  the  arc  of 

a  circle.  Then,  with  F  as  a  centre,  and  NH  as  a  radius, 
describe  an  arc  intersecting  the  arc  before  described,  at 
p  and  q  ;  these  will  be  points  of  the  hyperbola  ;  for, 
Fq  —  Fq^  is  equal  to  the  transverse  axis  AB. 

If,  -rt-ith  jP  as  a  centre,  and  F'H  as  a  radius,  an  arc  be 
described,  and  a  second  arc  be  described,  with  F'  as  a 
centre,  and  NH  as  a  radius,  two  points  in  the  other  branch 
of  the  curve  will  be  determined.  Hence,  by  changing  the 
centres,  each  pair  of  radii  wiU  determine  two  points  in  each 
brunch. 


I 


Third— When  the  axes  are  given. 

7.  Since  the  square  ot  the  semi-conjugate  axis  is  equal 
to  the  square  of  the  distance  from  the  centre  to  the  focus 
minus  the  square  of  the  semi-transverse  axis,  the  square  of 
the  distance  from  the  centre  to  the  focus  (Art.  5),  is  equal 
to  the  sum  of  the  squares  of  the  semi-axes. 

Let  AB  and  DB\  be  the  axes 
of  an  hyperbola. 

At  the  vertex  B^  draw  BH 
perpendicular  to  AB^  and  make 
it  equal  to  the  semi-conjugate 
axis  (7X>,  or  CD',  Join  H  and 
the    centre    C,     Then,    with    C 


BOOK   v.] 


THE      HYPERBOLA 


133 


as  a  centre,  and  CH  as  a  radius,  desciibe  a  semi-circum- 
ference, intersecting  AB  produced  in  F  and  F'  ;  these 
points  will  be   the   foci. 

The   curves  may  then  be  described  as  before. 


y 


Equation   of  the   Hjrperbola. 


8.  Let  F  and  F\  be  the 
foci,  and  denote  the  distance 
between  them  by  2c.  Denote 
the  semi-transverse  axis  CB^  by 
-4,  and  the  semi-conjugate,  (7Z>, 
by  B.  Let  P  be  any  point  of 
the  curve,  and  designate  the 
distance  F' P,  by  r\  and  FP^ 

by  r\   then,   2^1  will  denote  the  given  line  AB^  to  which 
the   difference,    F'P  —  PF^    is  to  be   equal. 

Through  the  centre  C,  draw  CD  perpendicular  to  F'F^ 
and  let  C  be  the  origin  of  a  system  of  rectangular  co-or- 
dinates. Let  X  and  y  denote  the  co-ordinates  of  any  point, 
as  P. 

The  square  of  the  distance  between  any  two  points  of 
which  the  co-ordinates  are  a;,  y,  and  x\  y'  (Bk.  I.,  Art.  19), 

(y  -  y'Y  +  (^  -  a')'. 

Jf  the  distance  be  estimated  from  the  point  F\  of  w^hich 
the   co-ordinates  are,   cc'  =  —  c,  y'  =  0,  we  shall  have. 


F'P    =  r'2  =  2/2  _p  (aj  ^  c)' 


(1.) 


and  if  it  be   estimated   from  the  point  F^   of  which    the 
co-ordinates  are    cc'  =  -f  c,    and    y'  =  0,    we  shall  have, 


134 


ANALYTICAL      GEOMETRY, 


[book  V. 


FP^  -  r^=  y^+  {x-cy.  .  (2.) 

If  we  add  and  subtract  Equa- 
tions  (1)   and   (2),  we   obtain, 

r'2  +  7-2  =  2(y2  +  iB2  +  c2). .  (3.) 

and,  r'2  —  r2  =  4cx  .  ,  (4.) 

Equation  (4)  may  be  put  under  the  form, 

{r'  +  r)  {r'  —  r)  =  4cx  .    .     .    .     (5.) 
But  we  have,  from  the  property  of  the  hyperbola, 

r'—r  =  2A     .....     (6.) 
Combining   (5)   and   (6),   we  have, 

.....    (7.) 


Combining   { 6 )   and   ( 7 ),  by  addition  and  subtraction,  we 
obtain, 

(8.) 


r'=A  +  ^ 


... 


and, 


r  =  — ^  + 


ex 


(9.) 


Squaring  both  members  of  Equations  ( 8 )  and  ( 9 ),  com- 
bining  the  resulting  equations,  and  substituting  the  values 
of  r'2  and  r^,  in  Equation  (3),  we  have. 


^2-H 


c^x^ 


2/2  +  a;2  4-  c^ 


BOOK  v.]  THE      HYPERBOLA.  135 

Substituting  for  c^,  its  value,    A^  +  JP   (Art.  7),  we  have, 
A^  4-  AW  +  ^23.2  ^  ^2y2  _|_  ^2a;2  +  ^*  +  A^B^ ; 

whence,  ^V^  -  -^^^^  =   —  ^^^^ 

which  is   the   equation    of  the    hyperbola,   refen-ed    to    its 
centre  and  axes. 


^^ 


Interpretation    of  the    equation. 


9. — 1.    If  in  the   equation   of  the  hyperbola, 

^2y2  _   ^2^2    _     _    J^2J^2^ 

we  make  y  =  0,  the  corresponding  values  of  x  will  be 
the  abscissas  of  the  points  in  which  the  curve  intersects 
the  axis  of  X  (Bk.  11.,  Art.  4—1);  viz.: 

X  =  -\-  A,    for    B;      and,      x  =   —  A,  for  A. 

2.  If  we  make  x  =  0,  the  corresponding  values  of  y 
will  be  the  ordinates  of  the  points  in  which  the  curve 
intersects  the   axis   of  Y^   viz. : 

y  =z   +  ^/^,    for  D;     and,     y  =   -  J? /^",  for  D' ; 

and  since  these  values  are  both  imaginary,  the  curve  does 
not  intersect   the   conjugate  axis  (Introduction,  p.  24). 

3.  If  we  place  the  equation  of  the  hyperbola  under 
the  form, 

J3   , 

y  =  ±  ~jv^^  ~  ^^^    ^^®  ^^®» 

1st.    That,  for   every  value   of  ar  <  A,   whether  positive 


136  ANALYTICAL      GEOMETRY.  [bOOK    V. 

or  negative,   tlie    corresponding    values    of  y  will    be    im- 
aginary. 

2d.  That,  for  every  value  of  ic  >  ^1,  whether  positive 
or  negative,  there  will  be  two  values  of  y,  numerically 
equal,    with   contrary   gigns. 

Hence,  we  see,  1st.  That  both  branches  of  the  curve 
are  symmetrical  with  respect  to  the  axis  of  X.  2d.  That 
they  do  not  approach  nearer  to  the  centre,  than  the  ver- 
tices B  and  A,  3d.  That  from  the  vertices,  they  extend 
indefinitely  in   the   direction  of  x  positive  and  x  negative. 

4.  By  a  course  of  reasoning  similar  to  that  pursued  in 
(Bk.  III.,  Art.  4 — 5),  we  find  the  following  analytical  con- 
ditions for  determining  the  position  of  a  point  with  respect 
to  the   hyperbola: 

Without  the  curve,        ^2^2  _  jpy.%  _}_  j\2jp.  y  q^ 
In  the  curve,  A^y"^  —  B'^x^  +  A'^B'^  —  0. 

Within  the  curve,  -4^  __  J522.2  ^  ^2^52  ^  q. 

5.  By  comparing  the  equation  of  the  hyperbola  with  the 
equation  of  the  ellipse,  referred  to  its  centre  and  axes,  it  is 
seen,  that  the  two  are  identical.,  except  in  the  sign  of  ^, 
which  is  minus  in  the  hyperbola,  and  plus  in  the  ellipse. 
We  may,  therefore,  pass  from  one  equation  to  the  other, 
by   substituting  for  B^    B^/—  1.     Hence,   it    follows,   that 

Every  result  obtained  from  the  equation  of  the  ellipse,  will 
become  a  corresponding  result  from  the  equation  of  the 
hyperbola,   by  changing  B    into  B^—  1. 


BOOK   v.] 


THE      HTPEKBOLA. 


137 


Equation  when  the  origin  is  at  either  vertex  of  the  transverse  axis. 

10.  If  ^Ye  transfer  the  origin 
of  co-ordinates  from  the  centre  C 
to  A,  one  extremity  of  the  trans- 
verse axis,  the  equations  of  trans- 
formation (Bk.  I.,  Art.  2§),  will 
reduce  to, 

X  =  -  A  -\-  x\  y  =  2/'. 

Substituting  these  values  in  the  equation  of  the  hyper- 
bola, it  reduces  to, 


Ay^  -  i?2ic'2  +  2B^Ax' 
which  may  be  put  under  the  form. 


0; 


A' 


y"  =  -ji[^" 


2Ax% 


which  is  the  equation  of  the  hyperbola,  referred  to  the  ver- 
tex A^  as  an  origin  of  co-ordinates. 

K  we  refer  it  to  the  vertex  ^,  as  an  origin,  the  equation 
■vsill  become. 


B^ 


y'2  =  jrpAx^  -f  a;'2). 


>< 


Conjugate  and  equilateral  hyperbolas. 

11.    If  on  the   conjugate  axis 
DD\  as  a  transverse,  and  a  focal 


distance  equal  to  ^/A^  -f  jB^,  Tve 
construct  the  two  branches  of  an 
hyperbola,  their  equation  will   be, 

j52^2  _  ^2^2  _  ^AW; 


138 


ANALYTICAL      GEOMETRY 


[book  V. 


or. 


^2y2   _   J52^2    ^    J[2^^ 


in  whicli  JB  denotes  the  semi-trans- 
verse axis,  and  A  the  semi-conju- 
gate. T^o  hyperbolas,  thus  con- 
nected, are  called  conjugate  hyper- 
bolas. The  conjugate  axis  of  the 
one,  is  the  transverse  axis  of  the 
other,  and  the  focal  distances  are 
equal. 

1.  If  the  transverse  and  conjugate  axes  are  equal,  the 
hyperbolas  are  called  Equilateral.  The  equation  then  be- 
comes, 

2/2  _  a.2  _    _  j2^     when  A  is  the  transverse  axis, 


and 


y 


JB\     w-hen  £  is  the  transverse  axis. 


These  correspond  to  the  case  in  which  the  ellipse  becomes 
the  circle. 

Eccentricity. 

12.  The  EccENTKicrrY  of  the  hyperbola,  is  the  distance 
from  the  centre  to  either  focus,  divided  by  the  semi-trans- 
verse axis,  and  is  denoted  by  e;  hence,    c  =  Ae. 


Polar  Equation- 

13.    Resuming    Equations    (8)   and    (9),    (Art.  8),   we 
have, 

»•'  =  ^  +  ^,  and     r  =  -  J.  4-  2  J 


or,    r'  =  A  -\-  ex,  ,  ,  (1.)     and     r  =  —  J.  +  «b  .  .  (2.) 


BOOK   v.]  THEHYPERBOLA.  139 

In  the  first  of  these  equations,  the  pole  lies  without  the 
curve,  and  in  the  second,  within  it. 

1.  If  the  origin  of  co-ordinates  be  transferred  to  the 
focus  F\  whose  co-ordinates  are,  —  c  =  —  Ae^  and  0, 
we  have  the  formulas   (Bk.  I.,  Art.  2§), 

X  —  —  Ae  +  x',        and        y  =  2/'* 

Substituting  this  value  of  cc,  in  Equation  (1),  and 
r'cosv,  for  x\  we  have, 

r'  =  A  —  Ae^  +  er'  cos  v ; 

Ail  —  e^) 

whence,  r'  =    -— ^ ^  , (3.) 

1  —  e  cos  V 

which  is  the  polar  equation  of  a  branch  of  the  hyperbola, 
in  terms  of  the  eccentricity  and  variable  angle,  when  the 
pole  is  without  the  curve. 

2.  If  the  origin  of  co-ordinates  be  transferred  to  the  focus 
^  whose  co-ordinates  are,  -f  c  =  Ae,  and  0,  (Bk.  I., 
Art.  28),   we  have  the  formulas, 

X  =  Ae  +  x\        and        y  =  2/'- 

Substituting  this  value  of  a;,  in  Equation  (2),  and  rcosv, 
for  x'j  we  have, 

r  =   —  A  -{-  Ae^  -f  er  cos  v  ; 

•• 

whence,  r  =  —  -— ^ , (4.^ 

1  —  e  cos  V  ^     ' 

which  is  the  polar  equation  of  a  branch  of  the  hyperbola, 
in  terms  of  the  eccentricity  and  variable  angle,  when  the 
pole  is  within   the  curve. 


14:0 


XALYTICAL      GEOMETRY, 


[book  V. 


3.  "We  sec,  from  Equations  (1)  and  (2),  that  the  values 
of  '»•'  and  r,  are  expressed,  rationally,  in  terms  of  the  ab- 
scissas of  the  points  in  which  the  radius-vector  intersects 
tJie  curve.  This  property  is  peculiar  to  the  focus,  as  a 
pole. 

Diameters. 

14.  A  Diameter  of  any  hyperbola  is  a  line  drawn 
through  the  centre  and  limited  by  the  curve.  The  points 
in  which  it  intersects  the  curve,  are  called  vertices  of  the 
diameter. 


Every    diameter   is    bisected    at  the    centre. 


15.  If,  in  the  equations  ex- 
pressing the  values  of  x',  y\  x'\ 
y"  (Bk.  m.,  Art.  8),  we  substi- 
tute for  B,    B^  —  1,    we  have, 


•='=  ^^\i^^^^    y'=   '-^^"vS^i 


■=-^^vkS'    y"=-^'"'\S^- 


Hence,  the  co-ordinates  of  H'  and  H^  are  equal,  with  con- 
trary  signs,   therefore,    CS  =  CH'. 


Relation    of   ordinates    to    each    other. 

16.    The  equation  of  the  hyperbola,  referred  to  the  ver 
tex  B  of  the  transverse  axis  (Art.  10),  is, 


BOOK   v.] 


THE      HYPERBOLA. 


141 


y. 


i?2 


If  we  designate  a  ^^articular 
ordinate  by  y',  and  its  abscissa 
by  x\  and  a  second  ordinate 
by  y",  and  its  abscissa  by  a;", 
"we  shall  have, 


^^ 


y'2  =  -^I'lAx'  +  cc'2),         and        y'"^  =  2^(2.iic"  +  a;"^). 
Dividing  the  first   equation  by  the  second,  we   obtain, 


{1A  +  x'  )x' 
(2^  +  x")x" 


or, 


,//2 


2/'^    :    y^    :  :     (2^  +  x')x'    :     {2A  +  a;")a;", 
in  which  the  segments,  are,  •» 

2A  4-  ic^     a;',         and         2A  +  a;",     a;". 

Hence,  the  squares  of  the  ordinates   are   to  each  other  as 
the  rectangles  of  the  segments. 

Parameter. 

ly.  The  Paeameter  of  the  transverse  axis,  is  the  double 
ordinate  passing  through  the  focus. 

To  find  its  value,  let  us  take  the  Polar  Equation  (4) 
(Art.  13), 

I  —  e  cosy 

If  we  make  v  =  90°,  we  have  (Bk.  III.,  Art.  li), 


14:2 


ANALYTICAL     GEOMETRY. 


[book  V. 


Substituting  for  e^^  its  value 


A^      ' 


and  reducing, 


i?2  _,  ^  2^         4^ 

r  =  —r\        hence,        Parameter  =  —-r-  =   ——r  • 
A^  A  2A 

If  we  write  this  in  a  proportion,  we  have, 

2A    :    2B    :  :    2B    :    Parameter;        that  is, 

The  parameter  of  the  transverse  ojxis^  is  a  third  propor- 
tional to  the  transverse  axis  and  its  conjugate, 

1.    The  numerator,  in  the  value  of  r'  or  r  (Art.  13),  is 
equal  to   half  the   parameter. 


-i 


Equation  of  the  tangent.    Sub-tangent. 


18.  The  equation  of  a 
tangent  to  an  ellipse,  at  a 
point  whose  co-ordinates  are 
a",  y"  (Bk.  m.,  Art.  14),  is 

A'yy"  +  B'xx"  =  A^B^. 


This  will  become  the  equa- 
tion of  a  tangent  to  the  hyperbola,  if  we  substitute  for  jB, 
B'^—  1    (Art.  9 — 5) ;  this  substitution  gives, 

A^yy"  -  B^xx''  =  -  A^B^; 

which  is  the  equation  of  a  tangent  to  the  hyperbola, 

1.    I^  in  this  equation,  we  make  y  =  0,  we  have, 

CT=x  =  ~r 


BOOK  v.]  THE      HYPERBOLA. 

Subtracting  this  from    CB  =z  x"    we  obtain, 
TB  = ~. 


143 


•which  is  the  value  of  the  sub-tangent. 


Equation  of  the  normal.    Sub-normaL 

19.  The  equation  of  a 
normal  line  to  an  ellipse,  at 
a  point  whose  co-ordinates 
are  x'\  y"  (Bk.  HI,  Art. 
16),  is 


-^ 


y-y"=B^'i---")- 

This  becomes,  by  changing  the  sign  of  jB^, 


y-y"=  -  W'(=^  - '"")' 

which  is  the  equation  of  the  normal,  PK. 

1.    To  find  the  point  in  which  the  normal  intersects  fhe 
axis  of  -3r,  make  y  =  0,   and  we  have. 


cjsr  =  X 


A^A-  B^ 


x'\ 


and  by  eubtractuig  ic",  we  find  the  sub-normal, 
BN^  4-  —W-' 


Ui 


ANALYTICAL      GEOMETRY.  [bOOK    V. 


Tangent  bisects  the  angle  of  the  two  lines  drawn  to  the  focL 

20.  If  from  P,  any  point 
of  the  curve,  we  draw  two 
lines  to  the  foci  F'  and  I\ 
and  recollect  that  CF'  or 
CF  is  equal  to  c  —  Ae 
(Art.  13),  we  have,  by  using 

the  value   o^   CT  =    —   (Art.  18), 

X 


^'        Aex"-A^        A{€x"-A)^ 
and        TF=Ae---r,  =  -7, =  -^^>7 ; 


hence,         F'T   :     TF   :  :    ex"  +  A    :    ex"  -  A, 

By  referring  to  the  values  of  r'  and  r  (Art.  8),  and  re- 
memhering  that   -j  =  ^'   ^^  have, 

r'    ;    r    :  :    ex"  ■\-  A    :    ex"  -  A; 
hence,*  F'T    :     TF   :  :    r'    :    r. 

Therefore,  PT  bisects  the  angle  F'FF\   Hence, 

i^  a  ?jwe  he  drawn  tangent  to  an  hyperbola  at  any  point, 
and  two  lines  he  drawn  from  the  same  point  to  the  foci, 
the  lines   drawn  to   the  foci  xcill  malce  equal  angles  with 

the   tangent.  _____^ 

*  Leg.,  Bk.  II.     Prop.  4.     Cor.  f  Leg.,  Bk.  lY.     Prop.  17. 


BOOK  V.J 


THE      HYPERBOLA 


145 


Supplementary  Chords. 

ai.  The  equation  of  condition 
of  supplementary  chords,  in  the 
ellipse  (Bk.  in.,  Art.  20),  is, 


aa'  z 

=   - 

J52 

Substituting 

for 

^, 

B^ 

1, 

we  have, 

aa'  = 

A' 

ir^ 


1.  If  the  chords  are  drawn  to  any  point  P,  in  the  branch 
IIBP^  the  tangents  a  and  a'  will  be  both  positive ;  if 
drawn  to  a  point  in  the  other  branch,  they  will  both  be 
negative. 

2.  If  tne  hyperbola  is   equilateral,    A  =  B,    and  there 

will  result, 

aa'  =  1, 

wliich  shows,  that  the  sum  of  the  two  acute  angles  formed^ 
by  the  supplementary  chords  xoith  the  transverse  axis,  o?i 
the  same  side,  is  equal  to   90°.* 


Supplementary  Chords.    Tangent  and  diameter. 

22.  In  the  ellipse,  the  relation 
between  the  tangents  of  the  angles 
which  a  tangent  and  the  diameter 
passing  through  the  pomt  of  con- 
tact make  with  the  transverse 
axis,  is  expressed  by  the  equation 


*  Lcgendre,  Trig.  Art.  61. 


10 


146  A>'ALYTICAL      GEOMETRY.  [BOOK   V. 

(Bk.  m,  Art.  21), 


aa 


B^ 

^2*' 


hence,  in   the  hyperbola,  it   is. 


aa   = 


A' 


But  the  equation  of  condition  of  supplementary  chords,  is 


aa    = 


hence, 


aa'  =  aa' ;        that  is. 


If  one  chord  is  parallel  to  a  diameter,  the  other  wiU  be 
parallel  to  the  tangents  drawn  through  its  vertices. 


Construction  of  tangent  lines  to  the  hyi)erbola. 

23.     The  property  proved  in  Art.  22,  enables  us  to  draw 
a  tangent  to  an  hyperbola,  at  a  given  point  of  the  curve. 

Let  (7  be  the  centre  of  the  hy- 
perbola, AB  its  transverse  axis, 
and  P  the  given  point  of  the 
curve  at  which  the  tangent  is  to 
be  drawn. 

Through  P,  draw  the  serai- 
diameter   PC,    and    through   A, 

draw  the  supplementary  chord  AH,  parallel  to  it.  Then, 
draw  the  other  supplementary  chord  BIT,  and  through  P, 
draw  PT  parallel  to  BB:-,  then  will  l^T  bo  tho  tangent 
xequired. 


•-V 

H 

\^ 

■^ 

A 

c-r\B      ■ 

/ 

\ 

BOOK  v.] 


THE      IIYBEKBOLA. 


147 


24.  The  property  proved  in  Art.  20,  enables  us  to  draw 
a  tangent  to  an  hyperbola,  through  a  given  point  without 
the  curve. 

Let  H  be  the  given  point. 
"With  this  point  as  a  centre, 
and  IIF^  as  a  radius,  describe 
the  arc  of  a  circle.  With 
F\  as  a  centre,  and  a  radius 
equal  to  the  transverse  axis, 
describe  the  arc  of  a  circle 
intersecting  the  former  at  G 
and  6^'.    Draw  i^' 6r,  cutting 

the  curve  in  P.     Through  P,   draw  UPT^   and  it  wdll  be 
tangent  to  the  hyperbola  at  P. 

For,  if  we  draw  7£F,  IIG,  we  shaU  have  HF  =  JIG,  by 
construction ;  and  since  jP  is  a  point  of  the  hyperbola,  and 
F'G  equal  to  the  transverse  axis,  we  shall  have  PF  =  PG; 
hence,  PT  is  perpendicular  to  FG;*  and  since  the  triangle 
FGP  is  isosceles,  PT  will  bisect  the  angle  F'PF,  and 
will,  therefore,  be  tangent  to  the  hyperbola. 

1.  The  tw^o  arcs  described  with  the  centres  F'  and  JI, 
intersect  each  other  in  two  points,  G  and  G';  a,  line  may, 
therefore,  be  drawn  through  F'  and  either  of  these  points, 
thus  giving  two  points  of  tangency,   and  two  tangents. 


Conjugate  diameters. 

25,     Two  diameters  of  an  hyperbola  are  said  to  be  con- 
jugate to   each  other,  when   either  of  them  I'S  parallel  to 
the  two  tangents  drawn  through  the  vertices  of  the  other. 
*  Legendrc,  Bk.  I.     Prop.  16.   Cor. 


148  ANALYTICAL      GEOMETKY.  [bOOKV. 

26.  The   equation   of   condition   of   conjugate  diameters 
in  the  elKpse   (Bk.  III.,  Art.  27),  is, 

A^  sin  a  sin  a'  4-  j5^  cos  a  cos  a'  =  0. 

Hence,  for  the  hyperbola,  it  is, 

A^  sin  a  sin  a'  —  J52  cos  a  cos  a,'  =  0, 

or,  A^  tan  a  tan  aj  —  B^  =  0. 

Hyperbola  referred  to  its  centre  and  conjugate  diameters. 

27.  The  equation  of  the  hyperbola,  referred  to  its  centre 
and  axes,  is 

^2^2   _    ^.^2     _      _  ^2J52^ 

The    fonnulas   for    passing  from    rectangular   to    oblique 
co-ordinates,   the   origin  remaining  the   same,   are, 

X  =  a'  cos  a  +  2/'  cos  a',  y  =  x'  sina.  -\-  y'  sin  a'. 

Squaring  these  values  of  x  and  y,  and  substituting  in  the 
equation  of  the  hyperbola,  we  have, 

(.42sin2a'— ^2cos2a')2/'2+(.4=sin2a-J52cos2a)a;'2  I  _  _j^2j^ 
-\- 2  {A'^sia  asm  a'  — J3- cos  a  cos  ci')x'y'  )~ 

But  the  condition,  that  the  new  axes  shall  be  conjugate 
diameters,  gives, 

A^  sin  a  sin  a'  —  i?2  ^os  a  cos  a'  =  0  ; 

hence,  the  equation  reduces  to 

(J.2  8in2a'-^2cos2a')y^2  ^  (^2sui2a  _  J52cos2a)a;'2  _  _  A^B^, 

If  we  suppose,  in  succession,  y'  =0,  a'  =  0,  and  denote 


BOOK   v.]  THE      HYPERBOLA.  149 

by  A'  and  D\  the  corresponding  abscissas  and  ordinates,  we 
find, 

A'^  = —— ,        B"- 


A?  sin^a  —  IP-  cos^a  '  A?  sin^a'  —  IP  cos^a' 

If  ^'2  is  positive,  A'  will  be  real,  and  the  diameter  will 
intersect  the  curve.     Under  this  sujDposition,  we  shall  have, 

A?  sin^a  <  JS^  cos^a,         or,         tan  «  <  -7  • 
But,  tan  a  tan  a!  =    -j^     (Art.  26) ; 

hence,        tan  a'  >  —  ;        or,        A?  sin^a'  >  B"^  cos^a' ; 

hence,  B"^  will  be  negative. 

The  supposition,  therefore,  w^hich  renders  A'"^  jDOsitive,  or 
A'  real,  gives  B'"^  negative,  or  B'  imaginary;  which  shows 
that  only  one  of  the  diameters  intersects  the  curve.  At- 
tributing to  -B'2^  itg  proper  sign,  we  have, 

A'^  =  -.„  .  /  ^ — ^ ,         -B'^  = 


A"^  sin^a  —  J?2  cos^a  '  A"^  sm^cc'—B'^  cos^a' 

Findmg  the  values  of  the  denominators  in  these  equations, 
substituting  them  in  the  general  equation,  and  reducing,  we 
obtain, 

A'Y^  -    J?'2^'2    _     _  ^'2^'2  . 

or'  omitting  the  accents  of  x  and  y,  since  they  are  general 
variables,  we  obtain, 

which  is  the  equation  of  the  hyperbola,  referred  to  its  centre 
and  conjugate  diameters. 


150 


ANALYTICAL      GEOMETRY, 


[book  V. 


Interpretation. 

1.  We  have  already  seen 
(Art.  9—2),  that  when  the 
transverse  axis  A£  is  real, 
the  conjugate  axis  DD'  will 
be  imaginary,  and  recipro- 
cally; that  is,  the  two  axes 
will  not  intersect  the  same 
branch  of  the  hyi^erbola.  The 
last  proposition  proves  the  same  property  for  any  two  con- 
jugate diameters. 

If  then,  2A'  designates  the  diameter  IT'IT,  1B'  will 
designate  the  conjugate  diameter  G' G^  terminating  in  the 
conjugate  hyperbola  ;  and  each  will  be  parallel  to  the  two 
tangent  lines   drawn  through  the   vertices   of  the   other. 

If  B'  were  made  real,  A'  would  be  imaginary,  and  the 
equation  would  represent  the  cuiwes  FJDG^  F'D'G\ 

2.  The  equation  of  the  hyperbola,  referred  to  its  centre 
and  conjugate  diameters,  being  of  the  same  form  as  when 
referred  to  its  centre  and  axes,  it  follows,  that  every  value 
of  a;,  will  give  two  equal  values  of  y,  with  contrary  signs ; 
or,  if  B'  were  real,  every  value  of  y,  would  give  two  equal 
values  of  a*,  with  contrary  signs  ;  hence,  each  hyperbola  is 
symmetrical  with  respect  to  the  diameter  which  it  inter- 
sects; that  is. 

Either  diameter  bisects  all  chords  drawn  parallel  to  t/ic 
otha',  and  terminated  by  the  curve, 

3.  It  may  be  readily  shown,  that  the  squares  of  the  or- 
dinates  to   either   diameter,   are    proportional    to  the  rect- 


BOOK    v.]  THEHYPERBOLA.  151 

angles  of  the  corresponding  segments,  from  the  foot  of  the 
ordinates  respectively,  to  the  vertices  of  the  diameter. 

4.  The  equations  of  the  hyperbola  and  ellipse,  referred 
to  their  centres  and  conjugate  diameters,  are  identical^ 
except  in  the  sign  of  jB'^,  which  is  minus  in  the  hyper- 
bola, and  plus  in  the  ellipse.  We  may,  therefore,  pass 
from  one  equation  to  the  other,  by  substituting  for  B\ 
B'\^  —  1.  Hence,  it  follows,  that,  every  result  obtained 
from  the  equation  of  the  ellipse^  loill  become  a  correspond- 
ing result  from  the  equation  of  the  hyperbola^  by  changing 
B'  into  ^V  -  1- 

Relation  between  the  azes  and  the  conjugate  diameters. 
28.    In  the  ellipse  (Bk.  III.,  Art.  3l),  we  have, 

A^  tan  a  tan  a'  +  i?^  =  0 (1.) 

^'^'  sin  (a'- a)  =  ^^;  and,    .     .     .     (2.) 
A'  -^  B"  =  A'  -{-  B'     ...    (3.) 

J>y    substituting    for    B^    B  -y/"^^    and    for    B .  ^^/TTT, 
we  have,  for  the  hyberbola, 

-l'^  tan  a  tan  a'— J5' =  0     ....     (1.) 
A'B'  sin  (a'-  a)  =  AB     ,     .     .      (2.) 

and,  A"  -  B"  =  A'  -  B'      ,     (3.) 

*■ 

Interpretation. 

Equation    (2). — Construct   a   rectangle    on    the    axes,    and 

a   parallelogram    on    two    conjugate    diameters.     Di-aw    from 

G    a    perpendicular    to    CII;     this    perpendicular    will    be 

equal    to    B'   sin  (a'—  a).      Hence,    the    area    of    the    paral 


153 


ANALYTICAL      GEOMETRY, 


[book  V. 


lelograra  CGJPJI,  is  equal 
to  ^'i?'sm(a'_  a)  =  AB', 
hence,  tlie  whole  parallelo- 
gram is  equal  to  the  whole 
rectangle.  Therefore,  the 
parallelogram  formed  by 
drawing  tangents  at  the 
four  vertices  of  conjugate 
diameters^  is  equivalent  to 

the    rectangle  formed   by    drawing    tangents    through    the 
vertices  of  the  axes. 

2.     Equation  (3). — This  equation, 
or,  4^ '2  _  4J5'2  ^  4^2  _  4^2^ 


expresses  this  property:  TJie  difference  of  the  squares  of 
two  conjugate  diameters  is  equal  to  the  difference  of  the 
squares   of  the   axes. 

Hence,  there  can  be  no  equal  conjugate  diameters,  unless 
A  =  B\  in  which  case,  the  hyperbola  is  equilateral,  and 
then,  every  diameter  will  be  equal  to  its  conjugate. 


(A/^    The  hyperbola  referred  to  its  asymptotes. 

29.  The  Asymptotes  of  an  hyperbola,  are  the  diagonals 
of  the  rectangle  described  on  the  axes,  indefinitely  produced 
in  both  directions. 

Thus,  II' 11^  G'G^  are  as}'mptotes  of  the  hj'perbola 
whose  transverse   axis    is  AB,   and    also   of   the   conjugate 


BOOK   v.] 


THE      HYPERBOLA 


153 


hyperbola  wLose  transverse 
axis  is  DD', 

If  we  designate  the  angle 
estimated  from  CB  around 
to  Cn^  by  a,  and  the  angle 
BGG^  by  cfJ  \  or,  what  is 
equivalent,  designate  the 
angle  BGG\  by  -a', 


tan  a  = 
or, 

tan^a  = 


B_ 
A 

j52 


(1.) 


tan  a 


A^  sin^a— ^2  cos^a  =:  0.   (  3.) 


B^ 

A^' 


-A-^'y     or, 


tan^a'  = 

^2sinV— ^2cosV=0.  (4.) 


Equations  (3)  and  (4)  express  the  relations  between  the 
angles  which  the  asymptotes  form  with  the  transverse  axis. 
They  are  called,  equations  of  conditio^i, 

1.  If  the  hj-pcrbola  is  equilateral,  A  =  B,  and  Equor 
tions  (1)  and  (2),  give, 

tan  a  =r   1,  and  tan  a'  =   —  1 ; 

which  shows,  that  the  asymptotes  make  equal  angles  xcith 
the  transverse  axis — one  lying  in  the  first  and  tJiird  angles, 
<tnd  the  other  i?i  the  second  and  fourth^  and  that  they  are 
at  right  angles  to  each  other. 

2.  Since  the  sine  of  the  angle  at  the  base  is  equal  to 
the  perpendicular  divided  by  the  hypothenuse;  and  the 
cosine,  to  the  base  divided  by  the  hypothenuse,*  we  have, 

*  Legcndre,  Trigonometry,  Art.  S7. 


154  ANALYTICAL      GEOMETRY.  [bOOK   V. 

^  (1.)         sina^  =         ~  ^     -■     (2.) 


cos  a  = 


-—=£=.      (3.)  cosa.'  =z   -—£==.'     (4.) 

Equation  of  the  curve  referred  to  its  asymptotes. 

30.     The  equation  of  the  hyi^erbola,  referred  to  its  centre 
and  axes  is, 

^2^2   _    ^2a;2   ^     __   ^2^, 

The  formulas    for    passing    from    rectangular    to    obliquo 
co-ordinates,  the   origin  remaining  the   same,  are, 

X  —  x'  cos  a  +  y'  cos  a',  y  =  cc'  sin  a  -|-  y'  sin  a'. 

Substituting  these  values,   and  reducing,   we   obtain, 

( .12  sin'a  — J52cos2a')y '2  ^  (^2  sm^ct—JS^  cos^ol)x' 


4-  2(^-12  sin  a  sin  a'—  B^  cos 


2a-^COs2a)a;'2)    ^  _  ^,^ 
a  cosa')ic'y'         i 


The  equations  of  condition  (Art.  29),  reduce  the  co- 
efficients of  x'\  and  y'^,  to  0.  Multiplying  Equations  ( 1 ) 
and  (  2  )  (Art.  29 — 2),  and  (  3  )  and  ( 4  ),  and  reducing,  the 
coefficient  of  x'y',  becomes, 

4^2^2 


^2  4.  ^2 » 


hence,  the  equation  of  the  hyperbola,  referred  to  its  centre 
and  asymptotes,  is, 

^y  =  — I — > 


BOOK  v.] 

or,  by  putting  Jf,  for 


THE      HYPEKBOLA 
^2    +   JJ2 


155 


,    and  omitting  the  accents, 


xy  =  M. 


Interpretation  of  the  equation. 

31.  If  lines  be  dra^vn 
through  the  vertices  of  the 
axes,  they  will  form  the 
rhombus  AD'BD.  The  di- 
agonals (7P,  (7§,  of  the 
rectangles  described  on  the 
semi-axes,  are  equal  to  each 
other,  and  each  is  equal  to 
V!Z2~+TB2.    But  these  are 

also  equal  to  the  diagonals  BD^  BD\  and  each  pair 
mutually  bisect   each   other  at  H  and   N. 

Hence,   CH  =  lv^+~B'^,      and      (7iV  =  l^^A^+B^; 

therefore,  CH  X   CN  =  A^B^  ^ 

4 

If  we   designate  the   angle  included  between  the  as}Tnp- 
totes  by  /3,  we  shall  have, 

^  CB^  X   CJSTsm  [3  =  xi/  sin  (3; 

the  first  member  of  the  equation  is  equal  to  the  rhombus 
CHBlsf'^  the  second,  to  any  parallelogram,  as  CQMK^ 
whose  sides  are  denoted  by  x  and  y;  that  is, 

TTie  rhombus  described  on  the  abscissa  and  ordinate  of 
the  vertex  of  the  hyperbola,  is  equivalent  to  the  paralklO' 


15f>  ANALYTICAL      GEOMETRY.  [bOOKV. 

gram  described  on  the  abscissa  and  ordinate   of  any  point 
of  the  curve. 

1.  The  rhombus  CSBK^  described  on  the  abscissa  and 
ordinate  of  the  vertex  of  the  hyperbola,  is  called  the  power 
of  the  hyperbola.  It  is  one-eighth  of  the  rectangle  described 
on   the   axes. 

Asjrmptotes    approach   the    curve. 

32.  Let   us  resume  the   equation  of  Art.  30, 

xy  =z  M,        from  which,        y  =  — • 

Since  M  is  constant,  if  x  increases  continually,  y  will 
diminish,  and  if  x  becomes  infinite,  y  will  become  0 ;  hence, 
the  hyperbola  continually  approaches  the  asymptote,  and  as 
y  cannot  become  negative  so  long  as  x  is  positive,  it 
foUows  that  the  curve  T\ill  touch  the  asymptote  when  y 
is  0.  The  same  may  be  shown  with  respect  to  the  other 
asymptote.    Hence, 

The  asymp^totes  continually  approach  tne  hyperbola^ 
and  become  tangent  to  it  at  an  infinite  distance  from  the 
centre. 

Asymptote,    the    limit    of  tangents. 

33.  The  equation  of  the  tangent,  when  the  curve  is 
is  referred  to   its  centre  and  axes  (Art.  18),  is, 

A^yy"  _  B^-xx"  =   -  A^JBK 

If  we  make    y  =  0,     we  find, 

A^ 

25     =     ZT5 


BOOK  v.]  ALGEBRAIC      CUR  YES.  157 

which    is    the    distance    from    the    centre    to    the   point  in 
which  the  tangent  intersects   the   transverse   axis. 

If  x"  increases,  x  diminishes,  and  if  x"  be  made  infinite, 
X  will  be  equal  to  0;  that  is,  the  tangent  line  Avill  pass 
through  the  centre,  and  since  both  the  tangent  and  asymp- 
tote touch  the  curve  at  a  point  infinitely  distant  from  the 
centre,  they  will  coincide. 

1.  The  asymptotes  have  been  defined  as  the  diagonals, 
prolonged,  of  the  rectangle  described  on  the  axes.  It  is 
easily  proved,  that  they  are  also  the  common  diagonals 
of  all  'parallelograms  formed  hy  drawing  tangent  lines 
through  the  vertices  of  conjugate  diameters. 

ALaEBRAIC        CUEVES. 
Classification. 

34.  An  ALGEBRAIC  CURAT:  is  one  in  which  the  relation 
between  the  co-ordinates  of  all  its  points  are  expressed  only 
in  algebraic  terms. 

35.  We  have  seen  that  every  equation  of  the  first 
degree,  between  two  variables,  is  the  equation  of  a  straight 
line   (Bk.  L,  Art.  18). 

We  have  also  seen,  that  the  equations  of  the  circle,  the 
ellipse,  the  parabola,  and  the  hyperbola,  are  all  of  the 
second  degree;  and  analogy  would  lead  us  to  infer,  that 
every  equation  of  the  second  degree  heticeen  two  variables, 
represents  one  or  the  other  of  these  curves.  This  is  now 
to  be  proved  rigorously. 

36.  The  general  equation  of  the  second  degree  between 
two  variables,  is, 


A- 


158 


ANALYTICAL      GEOMETEY. 


[book  V, 


A%f  +  Bxy  -^  Cx^-^-  Dy  +  Ex  ■\-  F  =  'y    .    (i.) 

which  contains  the  first  and  second  powers  of  each  variable, 
their  product,  and  an  absolute  term,  F, 

The  coefficients,  A^  J5,  (7,  7>,  jEJ  and  F^  are  entirely 
independent  of  the  variables  y  and  x.  They  are  called  con- 
stants;  or  arbitrary  constants,  since  values  may  be  attri- 
buted to  them  at  pleasure. 

37.  Let  us  suppose  that  the  co-ordinate  axes  are  rect- 
angular; this  supposition  will  not  render  the  discussion, 
or  the  results,  less  general.  For,  if  the  co-ordinate  axes 
were  oblique,  we  might  readily  pass  to  a  system  of  rect- 
angular co-ordmates,  without  affecting  the  degree  of  the 
equation^  since  the  equations  for  transformation  are  always 
linear. 


Change  of  direction  of  the  axes. 

3§.  To  pass  from  a  system  of  rectangular  to  a  system 
of  oblique  co-ordinates,  the  origin  remaining  the  same,  we 
have  (Bk.  I.,  Art.  29), 

sc  =  tc'  cos  a  -}-  2/'  cos  a',  y  —  cc'  sin  a  -f  y'  sin  a'. 

Substituting  these  values  of  x  and  y,   in  Equation  (1),  we 
have. 


A  sin^cc' 

B  sina'cosa' 

C  cosV 


(2.) 

y'2  -\-  2 A  sin  a  sin  a' 

_S  sin  a  cos  a' 

B  sin  a'  cos  a 

2  C  cos  a  cos  ol' 


x'y'  -f  A  sin^a 

^  sin  a  cos  a 

C  cos2a 


«'» 


+  Z)  sina' 
JS'cosa' 


y'  -h  i>sina  \x'-\-F=^  0 
EcosoL  I 


BOOK  v.]  ALGEBRAIC      CURVES.  159 

Since  a  and  a'  are  entirely  arbitrary,  we  may  assign  U> 
either  of  tbera,  such  a  value  as  will  reduce  the  co-efficient 
of  x'y'  to  0.    This  supposition  gives, 

2  A  sin  a  sin  a' + J? (sin  a  cos  a'  +  sin  a'  cos  a)  -f  2  (7cos  a  cos  a'  zr  0. 

If  we  suppose,  a'  —  a  =  90°,  the  new  axes  will  be  rect- 
angular, and  we  shall  have, 

sin  a'  =  cos  a,         and         cos  a'  =   —  sin  a. 

Making  the  substitutions,  we  have, 

2^  sin  a  cos  a  +  ^(cos^a  —  sin^a)  —  2  (7 sin  a  cos  a   =   0; 

or,  (A  —  (7)2sina  cos  a  +  _B(cos%  —  sin^a)  =  0. 

But, 

2  sin  a  cos  a  =  sin  2a,         and  cos^a  —  sin^a   =   cos  2a;* 

hence,     .  (^  —  (7)  sin  2a  +  j5  cos  2a  =z  0. 

Dividing  both  members  of  the  equation,  by  cos  2a,  we 
have. 


tan  2a 


(3.) 


c 


Therefore,  when  the  new  axis  SZ\  makes  with  the  prim- 
itive axis  -Z",  an  angle  equal  to  half  the  angle  whose  tangent 

IS -j -^,  the  coefficient  of  ic'y'  will  reduce  to  0. 

Equation  (  2  )  will  then  take  the  form,  omitting  the  accents, 

A'y^  4-  (7V  +  Z>V  +  ^'x  +  J^  =  0    .     .     (4.) 
*  Legendre,  Trig.  A.rt.  66. 


160  ANALYTICAL      G  E  O  M  E  T  K  Y.  [bOOK    V. 

Change  of  the  origin  of  co-ordinates. 

39.     The  formulas  for  passing  from  a  system  of  co-ordi- 
nates to  a  parallel  system  (Bk.  I.,  Art.  28),  are, 

X  =  a  -\-  x\        and        y  =  5  -f  y'. 

Substituting  these  values  of  x  and  y,  iu  Equation  (4),  we 
have, 


Ay"'  +  C'x'-'  +  ^A'h 


y'  +  20 'a 


+  A'h''  +  C'a^  +  D'h  +  E'a  ■\-  F  =  0  ,    .     (5.) 

In  Equation  ( 5  ),  a  and  h  are  entirely  arbitrary.  If  we 
attribute  to  them  such  values  as  make, 

2A'b  +  ^'  =  0,      whence,      J  =_-_,..(  c.) 

and,    iG'a  -{-  E'  =z  0,      whence,      a  =  —  —j^,^    .   .   (7.) 

and  put,  -  {A'b''  +  Co'  -\- D'b  +  Ea  +  E)  =  F\ 
Equation  (5)  will  become,  omitting  the  accents, 

^y -f-  C'x^  =z  F' (8.) 

Interpretation. 

40. — 1.  The  ti*ansformation,  from  Equation  (1)  to  (4), 
is  always  possible.  For,  such  a  value  may  be  given  to  a  or 
a',  as  shall  render  the  coefficient  of  x'y\  in  Equation  (2), 
equal  to  0. 

2.    The   transformation,  from  Equation    (4)    to    (8),  is 


BOOK   V.J  ALGEBRAIC      CURVES.  161 

always  possible,  except  in  the  cases  when  b  and  o^  in  Equa- 
tions ( 6  )  and  (  7  ),  are  both  infinite,  or  vrhen  either  of  them 
is  infinite.  Under  the  first  supposition.  A'  =  0^  and  C  =  0, 
which  causes  the  second  powers  of  the  variables  to  disap- 
pear, in  Equation  (4),  and  the  equation  then  becomes, 

D'y   +  E'x  +  F  =  0, 

which  is  the  equation  of  a  straight  line   (Bk.  I.,  Art.  IS). 

K  only  one  of  them  is  infinite,  as  a,  for  example,  then, 
C  =  0,  and  Equation  ( 5  ),  after  making  this  supposition, 
takes  the  form, 

^y  -f  Ex  =  F". 

If  we  now  transfer  the  origin  of  co-ordinates  to  a  point 

on  the  axis  of  JT,  such  that, 

_  F" 

X  -  -^-x, 

we  shall  have,      A'y^  -\-E'{^,  -  x'\   =  F" ; 

E' 
or,  2/^  =    ■j,x\  or,         y^  =   ^px, 

which  is  the  equation  of  a  parabola. 
Every  curve,  denoted  by  an  equation  of  the  form, 

y  =  Ipx, 

in  which  n  is  any  positive  number,  except   1,  is  called  ai 
parabola. 

If  n  =  2,  we  have  the  common  parabola.  If  n  =  3,  tho 
cubic  parabola,  &c. 

3.    Let  us  interpret  Equation  (8), 

A' if  -j-  Cx''  =  F\ 
11 


162  ANALYTICAL      GEOMETRY.  [bOOK   V. 

When  A'^  C\  and  I^\  are  all  positive,  this  is  the  equation 
of  an  ellipse,  referred  to  its  centre  and  axes  (Bk.  HE.,  Art. 
8) ;  then.  A'  =  A\  C  =  ^,  and  JF"  =  A''B\  If  A'  =  C\ 
it  becomes  the   equation  of  the  circumference  of  a  circle. 

4.  1^  A'  is  negative,  and  C  and  F'  positive,  then,  by 
changing  ,the  signs  of  both  members, 

A'lf-  CV  =   -F\ 

which  is  the  equation  of  an  hyperbola  referred  to  its  centre 
and  axes  ;  then,  A'  =  A'',  C  =  J^\  and  F  =  A^B^  (Bk. 
lY.,  Art.  8). 

5.  If  JL'  is  negative,  €'  positive,  and  F'  negative,  then, 
by  changing  the  signs,  we  have, 

which  is  the  equation  of  a  conjugate  hj^erbola ;  C  =  J3\ 
A'  =  A^,  F  =  A^JB^,  and  2B  the  transverse  axis  (Bk. 
v..  Art.  11). 

6.  If,  in  Equation  (  2  ),  we  attribute  such  values  to  a  and 
a',  as  shall  reduce  the  coefficients  of  the  second  powers  of 
the  variables  to  0  ;  and  then  transfer  the  origin  of  co-ordi- 
nates, so  as  to  get  rid  of  the  first  powers  of  the  variables, 
the  equation  wWi  take  the  form, 

xy  =  M, 

which  is  the  equation  of  an  hyperbola,  referred  to  its  centre 
and  asymptotes  (Bk.  Y.,  Art.  30).     Hence, 

Every  equation  of  the  second  degree  between  two  varh- 
hles^  will,  under  every  hypothesis,  represent  either  a  cir- 
cle,  an  ellipse,   a  parabola,   or  an  hyperbola. 


BOOK  v.]  ALGEBRAIC      CURVES.  163 

41. — 1.  Lines  are  classed  into  orders,  according  to  the 
degree  of  their  equations. 

2.  Straight  lines  are  represented  by  equations  of  the 
first  degree,  between  two  variables,  and  are  called,  lines 
of  the  first  order, 

3.  The  circumference  of  the  circle,  the  ellipse,  the  para- 
bola, and  the  hyperbola,  are  represented  by  equations  of 
the  second  degree,  between  two  variables ;  hence,  they 
are  called,  lines  of  the  second  order, 

4.  And  lines  denoted  by  an  equation  of  the  third  de 
gree,  are  lines  of  the  third  order  ;  and  similarly,  for  the 
higher   degrees. 

Equations  when  the  origin  is  in  the  curve. 

42. — 1.  The  equation  of  the  circle,  when  the  origiB  is 
in  the  curve,  is, 

y^  =  2JRx  —  x\ 

2.  The  equation  of  the  ellipse,  when  the  origin  ia  at 
the  vertex  of  the  transverse   axis,   is, 

y^=  J(2^aj-c«2). 

3.  The  equation  of  the  parabola,  under  the  same  hy- 
pothesis, is, 

y^  =  2px. 

4.  The  equation   of  the  hyperbola  is, 

y^=  ~{2Ax-\-x% 


164  ANALYTICAL      GEOMETRY.  [bOOK   V. 

These   equations  may  all  be   put  under  the   form, 

in  which  m  is  the  parameter  of  the  curve,  and  n  the  square 
of  the  ratio  of  the  semi-axes.  In  the  circle  and  ellipse,  n 
is  negative ;  in  the  hyperbola  it  is  positive,  and  in  the 
parabola  it  is  0. 

2.  The  curves,  whose  properties  have  been  discussed  in 
the  last  four  books,  are  precisely  those  which  are  obtained 
by  intersecting  the  surface  of  a  cone  by  planes,  as  is  shown 
in  Bk.  VI.,  Art.  45—50.  For  this  reason  they  are  called, 
Conic  jSections. 


BOOK    YI. 


SPACE— POINT  AND  LINE — PLANE — SURFACES. 


a.  Space  is  indefinite  extension,  and  is  entirely  similar 
in  all  its  parts.  The  geometrical  magnitudes  are  portions 
of  space.  Their  absolute  places  cannot  be  determined, 
either  by  construction  or  by  the  algebraic  analysis,  since 
there  is  nothing  fixed  to  which  they  can  be  referred. 
Their  relative  positions  may,  however,  be  easily  found, 
and  these  enable  us  to  discuss  and  develop  their  pro- 
perties. 

2,  Thus  far,  the  analysis  has  been  limited  to  points 
and  lines  lying  in  the  same  plane.  These  have  been  re- 
ferred to  two  axes,  making  a  given  angle  with  each  other. 
The  analysis  is  now  to  be  extended  to  points  and  lines  in 
space,  which  will  be  referred  to  three  planes,  at  right  angles 
to  each  other. 

3.  Through  any  point,  as  -4,  conceive  a  horizontal  plane 
to  be  drawn.  Through  the 
same  point,  conceive  a  ver- 
ticai  plane,  ZAJ^,  to  be 
dra^\Ti :  this  is  the  plane  of 
the  paper  and  intersects  the 
horizontal  plane  in  the  line 
X'AX.  Through  the  same 
point  conceive  a  second  ver- 


z 

/ 

X 

/ 

A 

y 

Z' 

166 


ANALYTICAL      GEOMETBY, 


[book  VL 


tical  plane  to  be  drawn,  per- 
pendicular to  the  plane  ZAX. 
This  plane  will  intersect  the 
horizontal  plane  in  the  line 
YA  Y\  and  the  first  vertical, 
in  the  line  ZAZ'.  These 
three  planes  are  called,  co- 
ordinate planes 

4.  Since  the  co-ordinate  planes  are  respectively  at  right 
angles  to  each  other,  the  line  of  intersection  of  either  two 
will  be  perpendicular  to  the  third:  and  this  line  of  inter- 
section is  called  the  axis  of  that  plane  to  which  it  is  per- 
pendicular. 

For  example :  Z  is  the  axis  of  the  horizontal  plane  YX"; 
y;  the  axis  of  the  first  vertical  plane  ZAX\  and  -X"  the 
axis  of  the  second  vertical  plane  ZAY.  The  three  are 
called,  the  co-ordinate  axes,  and  their  point  of  intersection 
Ay  the  origin  of  co-ordinates. 

5.  The  co-ordinate  planes  are  supposed  to  be  indefinite, 
and  hence,  they  will  divide  all  space  into  eight  equal 
parts,  or  triedral  angles,  having  the  origin  A,  for  a 
common  vertex.  Four  of  these  angles  are  above  the  hori- 
zontal plan^  YAX^y  and  four  below  it.  They  are  thua 
designated 

iZAX"        is  called  the        1st  angle. 
YAX'  "  2d      " 


X'AY' 

T'AX 


3d      «« 

4th     «* 


BOOK  VI.]  CO-OEDINATE      PLANES.  167 

The  fifth  angle  is  directly  beneath  the  first,  the  sixth 
beneath  the  second,  the  seventh  beneath  the  third,  and 
the   eighth  beneath  the   fourth. 

This  manner  of  naming  the  angles  difiers  from  that 
adopted  in  the  plane,  where  the  first  angle  is  beyond  the 
axis  of  abscissas,  and  where  we  pass  round  from  the  right 
to  the  left ;  both  methods  are  now  too  well  established 
to  be  changed,  merely  for  the  purpose  of  producing  uni 
formity. 

6.  The  distance  of  any  point,  in  space,  from  either  of 
the  co-ordinate  planes,  is  estimated  on  the  axis  of  that 
plane,   or  on   a  line   parallel  to   the   axis. 

7.  If  from  any  point,  in  space,  a  line  be  drawn  per- 
pendicular to  either  of  the  co-ordinate  planes,  the  foot  of 
the  perpendicular  is  the  projection  of  the  point  on  that 
plane. 

8.  The  line  in  which  any  plane  intersects  either  of  the 
co-ordinate   planes,  is   called    its  trace  on  that  plane. 

9.  If,  through  a  straight  line,  in  space,  a  plane  be 
passed  perpendicular  to  either  of  the  co-ordmate  planes, 
its  trace  is  called,  the  projection  of  the  line  on  that  plane. 

10.  Let  us  suppose  that  we  know  the  distance  of  a 
poiJit  from  the  three   co-ordinate   planes,  viz. : 

from     TZ     =    a, 

from    ZX    =    b, 

from     YJl    =    c. 


168  ANALYTICAL      GEOilETEY  [liOOKVl. 

From  the  origin  A,  lay  off  on  the  axis  of  JT,  a  dist- 
ance Ap  =  a,  and  through 
p  pass  a  plane  parallel  to  the 
co-ordinate  plane  YZ.  Its 
traces  jt?P',  pP,  will  be  re- 
spectively parallel  to  the  axes 
Z  and  I"  Lay  off,  in  like 
manner,  on  the  axis  of  1%  a 
distance  Ap'=  J,  and  through 

p'  pass  a  plane  parallel  to  the  co-ordinate  plane  ZJC.  Its 
traces  jo'P,  /)'P"  will  be  parallel,  respectively,  to  the  axes 
-X'  and  Z.  Since  the  point  must  be  in  both  planes,  at 
the  same  time,  it  will  be  in  their  common  intersection, 
which  is   perpendicular   to   the   horizontal   plane   at   P. 

Lay  off,  from  the  origin  of  co-ordinates,  on  the  axis  of  Z, 
a  distance  Ap"  =  c,  and  through  />'',  pass  a  plane  parallel 
to  Y^:  its  traces  p"P\  p'^P"-,  ^iH  be  parallel,  respect- 
ively, to  the  axes  JST  and  Y^  and  the  point  in  w^hich  this 
2>lane  is  pierced  by  the  perpendicular  to  the  horizontal 
plane  at  P,  will  be  the  position  of  the  required  point. 
The  point  will,  therefore,  be  projected  on  the  first  vertical 
plane  ZX^  at  P',  and  on  the  second  vertical  plane  ZY,  Sit 
F".     Its  co-ordinates,  are  P/,  pP^  and  pP'. 

The  distances  of  a  point,  from  the  co-ordinate  planes,  are 
expressed,  algebraically,  by 

a  =  a,  y  =  h  z=:  c, 

and  since  these  conditions  determine  the  position  of  the 
point,  they  are  called,  the  equations  of  the  p^oint. 

Hence,  the  equations  of  a  p>oint  are  the  equations  which 
express  its  distances  from  the  three  co-ordinate  planes. 


BOOK    VI.]  CO-ORDINATE      PLANES.  169 

11.    Let  us  consider,  separately,  the   conditions  wliich  de- 
termine the  distances  of  a  point  from  the  co-ordinate  planes. 

The  conditions,  a  =  ±  a, 

limit  the  point  to  one  of  two  planes  drawn  parallel  to  the 
co-ordinate  plane  YZ^  on  different  sides  of  the  origin,  and 
at  a  distance  from  it  equal  to  a. 

The  conditions,  y  =  zh  b, 

limit  the  point  to  one  of  two  planes  drawn  parallel  to  the 
co-ordinate  plane  ZJT,  on  different  sides  of  the  origin,  and 
at  a  distance  from  it  equal  to  b. 

If  these  conditions  exist  together,  the  point  will  be  lim- 
ited to  four  straight  lines,  parallel  to  the  axis  of  Z. 

The  conditions,  2  =  ±  c, 

limit  the  position  of  the  point  to  one  of  two  planes  drawn 
parallel  to  the  co-ordinate  plane  YJT,  on  different  sides  of 
the  origin,  and  at  a  distance  from  it  equal  to  c. 

If  all  the  conditions  exist  together,  the  point  will  be  either 
one  of  the  eight  points  in  which  the  two  last  planes  are 
pierced  by  the  four  parallels  before  drawn ;  and  each  of 
these  points  will  be  found  in  one  of  the  eight  angles,  formed 
by  the  co-ordinate  planes.  By  attributing  to  the  co-ordinates 
of  these  points  the  signs  plus  and  minus,  the  position  of 
any  one  of  them  may  be  exactly  determined.     Thus, 

1st     angle,  x  =z   -{-  a.        y  =   -\-  b^        z  =   -\-  Cy 

2d      angle,  x  =   —  a,        y  z=   +  b,        z  =   +  c, 

3d      angle,  x  z=   —  a,        y  =   -  b,        z  =   +  c, 

4th    angle,         x  =   +  a,        y  =   —  b,        z  =  -{-  e, 
8 


170  ANALYTICAL      GEOMETRY.  [bOOK   VI. 

5th  angle,  x  =   -{■  a,  2/=+  5,  2=— c, 

6th  angle,  x  =   —  a,  y  =   ■\-  h^  z  =  —  c, 

7th  angle,  x  =  —  a,  y  =   —  b,  z  =  —  c, 

8th  angle,  jc=+a,  y  =z  —  h^  z  =   —  c, 

12.  Smce  either  co-ordinate  denotes  the  distance  of  a 
point  fi'om  a  co-ordinate  plane,  it  follows,  that  when  this 
distance  is   0,  the   point  will  be  found  in  the  plane. 

Hence,  we  have  the  following  for  the  equations  of  the 
co-ordinate  planes  : 

For  the  co-ordinate  plane   YAX^  whose  axis  is  Z^ 

2  =  0,  X    and    y    indeterminate ; 

that  is,  X  and  y  must  be  indeterminate,  in  order  that  they 
may  represent  the  co-ordinates  of  every  point  of  the  plane. 

For  the  co-ordinate  plane  JCAZ^  whose  axis  is   YJ 

y  =  0,  X    and    z    indeterminate. 

For  the  co-ordinate  plane   YAZ^  whose  axis  is  X^ 
X  =  0^  y    and    z    indctenninate. 

13.  Since  either  axis  Hes  in  two  of  the  co-ordinate 
planes,  we   shall  have,  for   the    equations  of  the  axis  of  JC^ 

2/  =  0,        s  =  0,         and        x    indeterminate. 

For  the  equations  of  the  axis  of  !FJ 

as  =  0,        s  =  0,        and        y    indetermmate. 


BOOK   VI.]  LINES      IN      SPACE. 

For  the  equations  of  the  axis  of  Zy 

a;  =  0,        2/  ==  0,        and        z    indeterminate. 
And  for  the  origin,  which  lies  in  the  three  axes, 

ic  =  0,         y  =   0,         and         2  =  0. 
14.    We  also  have,  for  a  point  in  the  axis  of  X^ 

y  =  0,        s  =  0,        and        jb  =   ±  a. 
For  a  point  in  the  axis  of  YJ 

a;  =  0,        2  =  0,        and        y  =   db  5. 
For  a  point  in  the  axis  of  Z^ 

a;  =  0,         2/  =  0,         and        2  =    ±  c. 


171 


Distance  between  two  points. 

15.  Let  (ft  Q\  Q",)  be 
one  of  the  points,  and  (P, 
r,  P",)  the  other. 

Denote  the  co-ordinates  of 
the  first  point,  by  x\  y\  z\ 
those  of  the  second  by  x'\ 
y'\  z'\   and    the    length    of 
the  required  distance,  by  D, 
The  line   §P,  is  the  projec- 
tion of  the  given  line  on  the  co-ordinate  plane  of  YX^   Q' P 
Its   projection   on  ZX,   and  F"  Q"  its  projection   on    YZ , 
The  distance  i>,  will  be  the  hypothenuse  of  a  triangle,  of 
which  the  base  is  §P,  and  altitude  p'F\ 


172  ANALYTICAL      GEOMETRY.  [bOOKVI. 

But,    Qp   ^x"-x\    Fp  =  y"-y\    and   p' T  =  z" "  z' . 
In  the  right-angled  triangle  QPp>i  ^^^  have, 

Qf=  {x"-xy+  {y"-y'y\ 
hence,  i>2  ^  ^^»  _  r^y  +  (y//  _  y')2  4.  ^^n  _  ^ry^ 

and,  D=   y^x"  -  x'Y  +  (y"  -  y')'  +  (2"  -  ^Y- 

1.  The  projection  of  a  line,  on  either  of  the  co-ordinate 
axes^  is  that  part  of  the  axis  intercepted  between  the  two 
perpendiculars  drawn  through  its  extremities.  Hence,  if 
the  line  whose  length  is  2>,  be  projected  on  the  three  co- 
ordinate axes,  x"  —  x\  y"  —  y\  z"  —  z\  will  represent, 
respectively,  the  length  of  the  projection  on  each  axis; 
hence,  it  follows,  that  the  square  of  any  line  in  space^  is 
equal  to  the  sum  of  tJie  squares  of  its  three  projections  on 
the  co-ordbiate  axes. 

2.  If  one  of  the  points,  the 
one,  for  example,  of  which  the 
co-ordinates  arc  x\  y\  z\  be 
placed  at  the  origin,-  we  shall 
have. 


which  expresses  the  distance  from  the  origin  of  co-ordinates 
to  any  point  in  space. 


Line  and  co-ordinate  axes. 


16.     The  three  lines  Fp^  Fp\  Pp"  drawn  perpendicular 
to  the   co-ordinate   planes,  may  be  regarded  a?  the  three 


BOOK  VI.]  LINES      IN      SPACE.  173 

edges  of  a  parallelopipedon,  of  which  the  Ime  drawn  to  the 
origin  is  the  diagonal.  We  have,  therefore,  verified  a  pro- 
position of  geometry,  viz. :  the  sum  of  the  squares  of  the 
three  edges  of  a  rectangular  parallelopipedon  is  equal  to 
the  square  of  its  diagonal. 

1.  This  last  result  offers  an  easy  method  of  determining 
a  relation  that  exists  between  the  cosines  of  the  angles 
which  a  straight  line   makes  with  the   co-ordinate   axes. 

Let  US  designate  the  length  of  the  line,  passing  through 
the  origin  of  co-ordinates  by  r,  and  the  angles  which  it  forms 
with  the  axes,  respectively,  by  ^,  Y,  and  Z. 

We  shall  then  have  for  the  lines  Ap,  Ap\  Ap'\  which 
are  respectively  designated  by  x'\  y'\  s",  the  following 
values,  viz.: 

•  x"  =z  r  cos  JT,         y"  =  r  cos  rj         2"  =  r  cos  Z, 

By  squaring  these  equations,  and  adding,  we  obtain, 

rf.fn  4_  y"2  4.  ^"2  _  r2(cos'-X  +  cos2  T  -f  cos^Z). 

But  we  have   already  found, 

Hence,  cos^X-f  cos^F-f  cos^Z  =  1 ; 

that  LS,  the  sum  of  the  squares  of  the  cosines  of  the  three 
angles  which  a  straight  line  forms  with  the  three  co-or- 
dinate axes  J  is  equal  to  radius  square,   or  1. 

Equation    of  a  straight   line   in   space. 
IT.    Let  C'P'  be  the    projection  of  a  straight   line  on 


174: 


ANALYTICAL      GEOMETKT. 


[book  VI. 


the  co-ordinate  plane  ZX,  and  C'P",  its  projection  on 
the  co-ordinate  plane  YZ. 
Since  C'P'  is  the  pro- 
jection of  the  line  on  the 
co-ordinate  plane  ZX,  the 
line  itself,  in  space,  is  in 
the  plane  passing  through 
C  'P,'  and  perpendicular 
to  the  co-ordinate  plane, 
ZX  (Art.  9).  Since  C"P" 

is  the  projection  of  the  line  on  the  co-ordinate  plane  ZY^ 
the  line  itself,  in  space,  is  in  the  plane  passing  through  C"F" 
and  perpendicular  to  the  co-ordinate  plane  YZ  ;  hence,  it 
must  be  the  common  intersection  of  these  two  projecting 
planes.  The  conditions,  therefore,  which  fix  the  projections 
of  a  line,  will   determine  the    line  in  space. 

Let  a  =  a2  -|-  a, 

be  the  equation  of  the  projection   C'P',   and 
y  =  bz  i-  (3, 

the   equation  of  the  projection    C"P", 

In  these  equations,  a  denotes  the  tangent  of  the  angle 
ADP',  a  the  distance  AC',  b  the  tangent  of  the  angle 
JP"FZ,  and  /3  the  distance  AC".  The  angles  in  the  co- 
ordinate plane  ZX,  are  estimated  from  the  axis  Z  to  the 
right,  and  in  the  co-ordinate  plane  YZ,  'they  are  estimated 
from  the   axis  Z,  towards  the  left. 

If  we  suppose  a,  a,  b,  and  /3,  to  be  given,  the  two  pro- 
jections C'JP'i  C"F",  will  be  determined;  and  hence,  the 


BOOK   VI.] 


LINES      IN      SPACE. 


175 


line,  of  which  they  are  the  projections,  will  he  determined 
in  space.     Hence, 


az  -f  ct, 


y  z=  bz  -h  (3, 


are  the   equations  of  a  straight  line. 

1.  Since  the  projections  of  a  straight  line  on  the  two 
co-ordinate  planes  Z^,  ZY,  determine  the  position  of  the 
line  in  space,  they  ought,  also,  to  determine  its  projection 
on  the  third  co-ordinate  plane,  YX^.  This  indeed  may  be 
easily    proved. 

For,  through  P'  draw  a 
parallel  to  the  axis  of  Z, 
and  from  the  point  in 
which  it  intersects  the  axis 
of  JT,  draw  a  parallel  to 
axis  of  Y.  Through  P" 
draw  a  parallel  to  the 
axis  of  Z,  and  through  the 
point  in  which  it  intersects 

the  axis  of  Y,  draw  a  parallel  to  the  axis  of  -Z";  then  will 
P  be  the  projection  of  the  point  (P,  -P"),  on  the  co-or- 
dinate plane   YJT. 

Find,  in  a  similar  manner,  the  projection  of  a  second 
point,   as  (C,   C"),  and  draw  the  projection    CP. 


V 

V 


Interpretation   of  the    equations    of  a    line. 

18. — 1.    Let  us  now  consider  the   equations, 

aj  =  az  -f  a,  y  =  ^2  -f  /3, 

separately. 


17G  ANALYTICAL      GEOMETRY.  [bOOK    VX 

The   equation, 

aj  =  as  +  a, 

being  independent  of  y,  will  be  satisfied  for  every  point  of 
the  plane  passing  through  C'F',  and  perpendicular  to  the 
co-ordinate  plane  ZX;  hence,  it  may  be  regarded  as  the 
equation  of  that  plane. 

In  like  manner,    the   equation, 

y  =  52  +  /3, 

being  independent  of  ic,  will  be  satisfied  for  every  point 
in  the  plane  passing  through  CF"^  and  perpendicular  to 
the  co-ordinate  plane  YZ\  hence,  it  may  be  regarded  aa 
the   equation   of  that  plane. 

2.  Let  us  now  consider  the  conditions  which  would  be 
unposed  upon  the  straight  line,  by  supposing  a,  5,  a,  and 
jS,   to  become  known,   in   succession. 

When  a,  J,  a,  and  ^,  are  aU  undetermined,  the  equations, 

JB  =  as  +  a,  2/  r=  J2  4-  /3, 

may  be  made  to  represent  every  straight  line  which  can  be 
drawn  in  space,  by  attributing  suitable  values  to  a,  J,  «, 
and  i3.  And  when  a,  J,  a,  and  ^,  have  given  values,  the 
equations  wiU   designate   but   a  single   straight  line. 

If  we  suppose  a  to  be  given,  the  line  may  have  any 
position  in  space,  such,  that  its  projection  on  the  co-or- 
dinate plane  ZX^  shall  make  an  angle  with  Z,  of  which 
the  tangent   is   a. 

If  we  suppose  a  also  to  be  given,  the  projection  of  the 
line  on  the  co-ordinate  plane  ZX^  will  intersect  the  axis 
of  -X"  at  a  given  point,  and  the  two  conditions,  wiU  limit 


UOOK   VI.J  LINES      IN      SPACE.  177 


the  line  to  a  given  plane.  Its  position  in  the  plane  will 
btill  be   entirely  undetermined. 

If  we  now  suppose  h  to  be  given,  the  direction  of 
the  line  will  then  be  determined,  but  it  may  still  have 
an  indefinite  number  of  parallel  positions  in  the  given 
plane. 

If  finally,  we  attribute  a  value  to  ^,  the  projection  on 
the  plane  of  YZ^  will  mtersect  the  axis  of  Y  at  a  given 
point ;  and  hence,  the  position  of  the  line  will  become 
known.  The  letters  a  and  S  represent  the  co-ordinates  of 
the  points  in  which  the  line  mtersects  the  co-orduiate  plane 

rx. 

The  resolution  of  problems  involving  the  straight  line 
in  space,  consists  in  finding  such  values  for  the  arbitrary 
constants  a,  5,  a,  and  /3,  as  shall  satisfy  the  required  con- 
ditions. 

Equations  of  a  line  passing  through  two  points. 

19.  Let  x\  y\  z\  and  a;",  y'\  2",  be  the  co-ordinates 
of  two  given  points. 

The  required  equations  will  be  of  the  form, 

a;  =  as  +  a    .   .    (1.)  y  =  ^»s  -f  /3   .    .    (2.) 

in  which  it  is  required  to  find  such  values  for  a,  a,  J,  and' 
/2,  as  shall  cause  the  right  line  to  fulfill  the  required  con- 
ditions. 

Since  the  straight  line  is  to  pass  through  a  point,  of  which 
the  co-ordinates  are  x\  y\  z\  we  shall  have, 

a;'  =  «2'  -i-  a   .    .    (3.)  y'  =  hz'  ^  f^    .    .    (4.) 

12 


178  ANALYTICAL      GEOMETEY.  [bOOK   \1, 

and  since  it  is  also  to  pass  through  a  point,  of  which  the 
co-ordinates  are  ic",  y",  s",  we  shall  likewise  have, 

x"  =  az"  4-  a   .   .   (5.)  y"  =  hz"  +  ^   .   .   (6.) 

The  last  four  equations  enable  us  to  determine  the  four 
constants,  a,  a,  J,  /3. 

By  subtracting  Equation  (  5  )  from  Equation  ( 3  ),  and  ( 6  ) 
from  (4),  we  obtain, 

X'  -  x"  =  a(z'  -  z'%        and        y'  -  y"  =  h{z'  -  z'% 

fi-om  which  we  find, 

^ --'--"     ana    j>  =  y^--y:^^ 


z'  -z'"         ^   ~    z'  -z"' 

hence,  a  and  h  are  detennined.  If  these  known  values  be 
substituted,  respectively,  in  Equations  (3)  and  (4),  or  in 
(5)  and  (6),  the  values  of  a  and  /3  will  become  known,  and 
either  set  would  represent  the  required  line. 

But  it  is  more  convenient  to  have  the  equations  under 
another  form.  Subtract  Equation  (  3 )  from  Equation  ( 1 ), 
and  (  4  )  from  ( 2 ) ;   we  then  have, 

X  —  x'  =  a{z  —  s'),  y  —  y'  =  b(z  —  z'), 

which  are  the  equations  of  a  straight  line  passing  through  a 
given  point.  Substituting  for  a  and  b,  their  known  values, 
we  have, 

■x-x'  =  J^-^;  {z  -  z'),        y-y'=  5--^'  {z  -  z\ 

which  are  the  equations  of  a  straight  line  passing  througli 
the  two  given  points. 


BOOK  VI.]  LINES      IN      SPACE.  179 

Lines  intersecting  and  paralleL 

20.     It    is  required    to  find    the    conditions    which  will 
cause  two  lines  to  intersect   each  other. 

Let  X  =  az  -{-  n,  y  =z  hz  -\-  ^,' 

and  X  =  a'z  -{■  a',  y  =  h'z  +  P\ 

be  the  equations  of  the  lines,  in  which  the  arbitrary  con- 
stants   a,  a,  b,  (3,    a\  a',  h\  /3',    are  undeteraiined, 

If  these  lines  intersect  each  other  in  space,  they  must 
have   one    point  in   common,   and   the  co-ordinates  of  this 
•  point  will  satisfy  the   equations  of  both  lines.     If  wc   de- 
signate the   co-ordinates  of  the  common    point  by    x\  y\ 
z\  we  shall  have, 

x'  =  az'  +  a    .   .    (1.)  y'  =  hz'  -\-  ^    .   .    (2.) 

x'  =  a'z'  -fa'..    (3.)  y'  =  h'z'  -f  /3'   .    .    (4.) 

Eliminating  x'  and  y'  from  these  equations,  we  find, 

{a  —  a')z'  -f  a  —  a'  =   0    .     .     .     .     (5.) 

(5_  j^y +/3  _/3'=  0  .    .    .    .     (6.) 

and  if  s'  be  eliminated  from  the  two  last  equations,  we  have, 

(a  -  a')  (/3  -  ^')  -  (a  -  a')  {h  -  h')  =  0, 

which  is  called    the  equation  of  condition,  since  it  must 
always  be  satisfied  in  order  that  the  two  straight  lines  may 
intersect  each  other. 
^  There  are    eight    arbitrary   constants    entering    into  this 


180  ANALYTICAL      GEOMETRY.  [bOOKVI. 

equation.  It  may,  therefore,  be  satisfied  in  an  infinite  num- 
ber of  ways.  Indeed,  if  values  be  attributed,  at  pleasure, 
to  seven  of  the  constants,  such  a  value  may,  in  general, 
be  found  for  the  remaining  one,  as  will  satisfy  the  equation, 
and,  consequently,  cause  the  lines  to  intersect  each  other. 


When  parallel. 

21.    Let  us  now  find  the   co-ordinates  of  the  point  of 
intersection.    "We  find,  from  Equations  (  5  )  and  (  6  ), 

,         a'  -a                                              /3'_^ 
12!  —    ;  ,  or  Z'  —    -J- jj  . 

a  —  a  0  —  0 

Substituting  this  value  of  2',  in  Equations  (1)  and  (2), 
we  have. 

These  values  of  the   co-ordinates  of  the  point  of  inter- 
section, become  infinite,  when 

a  =  a\        and        h  =  V  \ 

that  is,  when  the  projections  of  the  lines  on  the  co-ordi- 
nate planes  ZX  and  ZY,  are  parallel. 

1.    If  we  have,  at  the  same  time, 

a'  =  a,        and        /S'  =  /?, 

the  co-ordinates  of  the  point  of  intersection  will  become 
- ,  or  indeterminate ;  as,  indeed,  they  should  do,  since  the 
two  lines  would  then  coincide  throughout  their  whole  extent. 


BOOK  Vl.J 


LINES      IN      SPACE. 


181 


Angle  between  two  lines. 


22.     Let 


a;  =  as  +  a,  y  =  bz  -\-  (3^ 

be  the  equations  of  the  first  line,  and 

aj  =  a's  +  a',  f/  =  b'z-h  (3\ 

be  the  equations  of  the  second. 

It  has  been  shown,*  that  two  straight  lines  which  cross 
each  other  in  space,  may  be  regarded  as  forming  an  angle, 
although  they  do  not  lie  in  the  same  plane.  They  are  sup- 
posed to  make  the  same  angle  with  each  other  as  would  be 
formed  by  one  of  the  lines,  and  a  line  drawn  through  any 
point  of  it,  and  parallel  to  the  other ;  or,  as  would  be  formed 
by  two  lines  drawn  through  the  same  point,  and  respect- 
ively parallel  to  the  given  lines. 

If,  then,  two  lines  be  drawn 
through  the  origin  of  co-ordi- 
nates, respectively  parallel  to  the 
given  lines,  the  angle  which  they 
form  with  each  other  will  be 
equal  to  the  required  angle. 

The  equations  of  these  lines 
will  be, 

X  =  az,  y  =  hz^ 


X  =  a'z, 


for  the  first, 


b'z,       for  the  second. 


Let  us  take,  on  the  first  line,  any  point,  as  -P,  and  desig- 
nate its  co-ordinates  by  x\  y\  z\  and  its  distance  from  the 
origin,  by  r'.    Take,  in  like  manner,  on  the  second  line,  any 
*  Legendre,  Bk.  VL     Prop.  6.    Sch. 


182 


ANALYTICAL      GEOMETBY. 


point,  as  P",  and  designate  its 
co-ordinates  by  Jc",  y",  s",  and 
its  distance  from  the  origin,  by 
r',  and  let  D  denote  the  dis- 
tance between  the  points.  If 
we  designate  the  angle  included 
between  the  lines,  by  F",  we 
shall  have,  in  the  triangle  AP'P"^ 

cosF=  ^n , 


•    •    • 


(1) 


and  we  have  now  only  to  find  r\  r",  and  2>. 

Let  ns  designate  the  three  angles  which  the  first  line 
forms  with  the  co-ordinate  axes,  respectively,  by  -Z",  J^  and 
Z,  and  the  angles  which  the  second  line  forms  with  the 
8ame  axes,  by  X\  Y\  and  Z' ;  we  shall  then  have  (Art..  16), 

sc'   =  r'  cosJT,  y'  =  r'  cosYi  ^   =.  r'  cosZ, 

x"  =  r"cosX',  y"  =  r"cosF',  s"  =  r"cos^. 

But  the  square  of  the  distance  between  two  points  is, 

i)2  =  {x'  -  x"Y  +  (y'  -  y"Y  -f  {z'  -  z"Y  (Art.  15),  or, 
7>2  =  x"'-\-y'^+  z'24-  x"^-\-  2/"2+  z"^-  2{x'x"-^  y'y" ^  z'z")  ; 

or,  by  substituting  for  the  co-ordinates  of  the  points,  their 
diiitances  from  the  origin  into  the  cosines  of  the  angles 
which  the  lines  make  with  the  co-ordinate  axes,  we  have, 


r'=(cos'X+  cos'Y+cos^Z)  +  r"='(cos'X'  -f-  cos»r  +  cos'Z')] 


2r'r"(cosXcos^'  +  cosFcoaF'  +  cos  Z  cos  2') 


*  Leg.,  Mens.  Art.  97. 


BOOK  VI.]  LINES      IN      SPACE.  183 

But  it  has  been  shown  (Art.  16 — 1),  that, 

C0S2X+  C0S2  Y+  COS^Z  =   1 ,  C0S2  JP  +  0082!^  +  cossZ'  =  1 ; 

and  hence, 

j)2  _  ^2 1.//2_  2rV'XcosXcosX''+cosFcosF'+cosZcosZ'). 

If  this  value  of  J)^  be  substituted  in  Equation  (1), 

cosF= ^r, , 

we  shall  find,  after  dividing  by  2rV", 

cos  Y  =  cos  X  cos  X'  4-  cos  Y  cos  Y'  -\-  cos  Z  cos  Z' ; 

that  is,  the  cosine  of  the  angle  included  hetioeen  two  I'mes^ 
is  equal  to  the  sum  of  the  rectangles  of  the  cosines  of  the 
angles  which  the  lines  in  space  form  with  the  co-ordinate 
a'jr4^s. 

Angle  under  another  form. 

23.  Having  found  the  cosine  of  the  angle  included  be- 
tween two  lines,  in  terms  of  the  angles  which  they  form 
with  the  co-ordinate  axes  in  space,  we  shall,  in  the  next 
place,  find  the  same  value  in  terms  of  the  angles  which  the 
projections  of  the  lines  on  the  co-ordinate  planes  ZX  and 
YZ^  form  with  the  axis  of  Z. 

The  equations  of  the  parallel 
lines  drawn  throuc^h  the  orii^in, 
are, 

X  =  az^  y  •=  bz, 

X  =  a'z,          y  —  b'z,  y^ 

Let  us  desi ornate  the  co-ordi- 


I  Si  ANALYTICAL      GEOMETRY.  [bOOK   TT. 

natos  of  tlie  point  P,  on  the  first  line,  by  x\  y\  z' ;  we  shall 

then  have, 

ic'  =  az\  y'  =  hz'-, 

and  for  the  A'alue   of  r', 
From  these  three  equations  we  find, 


y'  = 


But  we  have  already  found  (Art.  16), 

x'  =  r'  cos  X,         y'  —  *''  cos  J^         z'  =  r'  cos  Z. 
Substituting  these  values,  and  dividing  by  r',  we  obtain, 

co&X=     .  :,  cos  JF—     . -,  cosZ=- 


'v/r-fa2_|_J2  yi+«2^j2  0_|_a2^J2 

If  we  reason,  in  the  same  manner,  on  the  equations   of 
the  second  line,  we  shall  find, 

a'  h'  1 

cos  A"  =       =:,     cosF'  =      ■ ; ,     cosZ' 


If  these  values  be  now  substituted  in  equation  for  cos  F", 
we  shall  have, 

1  4-  aa'  +  hh' 


cosT^  = 


±  ^/l~+~d'  +  62  yi  +  a""  +  h"" 


The  cos  "F  will  be  plus  or  minus,  according  as  we  take 
the  signs  of  the  radical  factors  in  the  denominator,  like  or 
unlike. 

The  plus  value  of  cos  Y  will  correspond  to  the  acute 
angle,  and   the   minus  value,  to   the   obtuse   angle. 


BOOK   VI.]  LINES      IN      SPACE.  185 

1.    If  we  make,     v  =  90°,     cos"F  =  0,     and 

1  +  aa'  +  bb'  =  0, 

which  is  the  equation  of  condition,  when  the  two  lines  are 
perpendicular  to  each  other  in  space. 

EXAMPLES. 

1.  What  is  the  distance  between  two  points  of  which 
the   equations  are, 

aj'  =  5,    y'  =  5,   z'  =  S;     x"  =  -  1,    y"  =0,    s"  =  -  5  ? 

Ans,    11.18  -f 

2.  Find  the  equations  of  a  line  which  shall  pass  through 
a  point  whose  co-ordinates  are,  a;'  =  3,  y'  =  —  2,  and 
z'  =  0,     and  be  parallel  to  a  line  whose  equations  are, 

X  =  z  -\-  1,     and    y  ■—  \z  —  2. 

A71S.     CC   =   2+3,     2/    =   -i2  —  2. 

3.  Required  the  equations  of  a  line  passing  through  the 
two   points  whose   co-ordinates  are, 

x'  =2,       y'   =   1,       ^'  =  0, 
and  aj"  =   -  3,       y"  =  0,       z"  =   —  1. 

A71S.    X  =  5z  +  2,     y  =  z  -{-  1. 

4.  Required  the  angle  included  between  two  lines,  whose 

equations  are, 

X  =  dz  -\-  5  )      p  ^,      ,  ^ 
{.   of  the  1st, 

y  =  5z  -i-  3) 
and  X  =     z  +  1 


of  the  2d. 
y  =  2z  ) 


Ati^.    14°  58'. 


186 


ANALYTICAL      GEOMETRY 


[cook  VI. 


5.    Required  the  angles  which  a  straight  line  makes  with 
the  oo-ordinate   axes,  its   equations  being, 


X  =    —22-1-1, 
y  =  s  +  3. 

Ans. 


"144°  44'  with  X, 
65°  54'  with  Y, 
65°  54'  with  Z, 


6.    Having  given  the   equations  of  two  straight  lines, 

2s+  1    ) 


X 


y  =z  22  +  2    f 


and 


X   =      2  +  5 

y  =  42  +  /3 


■1 


of  the  1st, 


of  the  2d, 


required  the   value   of  /3'   so   that  the  lines  shall  intersect 
each  other,  and  the  co-ordinates  of  the  point  of  intersection. 

/3'  =    -  6, 


Ans. 


OF        THE        PLANE. 


X'   = 


1  y'  =     10, 

U'    =         4. 


24,  The  EQUATION  of  a  plane  is  an  equation,  express- 
ing the  relation  between  the  co-ordinates  of  every  point  of 
the  plane. 

To    find    the    equation    of   a    plane. 

25.  A  line  is  said  to  be  perpendicular  to  a  plane  when 
it  is  perpendicular  to   every  line  passing  through  its  foot 


BOOK   VI.]  THE      PLANE.  187 

and  lying  in  the   plane:   and,   conversely,  the  plane  is  said 
to   be   perpendicular   to   the   line.* 

A  plane  may,  therefore,  be  generated  by  drawing  a  line 
perpendicular  to  a  given  line,  and  then  revolving  this  per- 
pendicular about  the  point  of  intersection.  K  the  perpen- 
dicular be  at  right  angles  to  the  given  line,  in  all  its  posi- 
tions, it  will  generate  a  plane  surface. 

Let  X  =  az  +  a,  y  =  52  -f  /5, 

be  the  equations  of  a  given  line. 

If  we  designate  the  co-ordinates  of  a  particular  point,  by 
x\  y\  z\  the  equations  of  the  line  passing  through  this 
point,   will  be, 

X  -x'  =  a{z  -  z')    .     (1.)         y  -  y'  =  h{z  -  z')    .     (2.) 

The  equations  of  a  second  line  passing  through  the  same 
point,   of  which  the  co-ordinates  are  x\  y\  z\   are  of  the 

form, 

X  —  x'  =  a' {z —  z')    .    .     .     .     (3.) 

y  -y'  =  b'{z-z')     .     .     .     .     (4.) 

But  the  two  lines  will  be  at  right  angles  to  each  other, 
if  their  equations  fulfill  the  condition  (Art.  23 — 1). 

I  +  aa'  +  bb'  =  0 (5.) 

^  K  we  now  attribute  to  a'  and  b\  all  possible  values  that 
will  satisfy  this  equation,  we  shall  have  all  the  perpen- 
diculars which  can  be  drawn  to  the  given  line,  through 
the  point  whose  co-ordinates  are  x\  y\  z' ,  These  perpen- 
diculars  determine  the   plane. 

*  Legendre,  Bk.  YI.  Def.  1. 


188  ANALYTICAL      GEOAIETRY.  [bOOKVI. 

It  id  necessary,  however,  to  find  the  equation  of  the 
plane  in  terms  of  the  co-ordinates  of  its  different  points 
MVe  find  from  Equations   ( 3 )   and   ( 4 ), 

z   —  z'^  z  —  z'  ' 

Substituting  these  values   in  Equation  (5), 

1  +  aa'  +  hb'  =  0, 

and  reducing,   we  find, 

z  —  z'  +  a{x  —  x')  +  b(y  -^  y')  =  0; 

but,  since  a,  b,  z\  x\  y\  are  known  quantities,  we  may 
denote  the  constant  part  of  the  equation  by  a  single  letter, 
by  making, 

—  z'  —  ax*  —  by'  —   —  c. 

hence,  the  equation  of  the  plane  becomes, 

z  -\-  ax  -\-  by  —  c  =  0. 

1.  Since  the  equation  of  a  plane  contains  three  vari- 
ables, we  may  assign  values,  at  pleasure,  to  two  of  them, 
and  the  equation  will  then  make  known  the  value  of  the 
third.  For  example,  if  we  assign  known  values,  denoted  by 
x'  and  y\    to  x  and  y,  the  equation  of  the  plane  will  give, 

z  =  c  —  ax'  —  by', 

and  hence,  the  co-ordinate  z  becomes  known. 

Traces    of   planes. 
26,    The  lines  in  which  a  plane   intersects  the   co-ordi- 


liOOK  VI.] 


THE      PLANE. 


189 


nate    planes,   are    called    the    traces  of  the  plane.      These 
traces  are  found  by  combining  the   equation  of  the  plan© 
with  the  equations  of  the   co-ordinate  planes. 
Thus,  if  in  the  equation, 

z  •\-  ax  +  hy  —  c  =  0, 

we  make  y  =  0,  which  is 
the  characteristic  of  the  co- 
ordinate plane  ZJT^  the  re- 
sulting equation, 

s  +  aic  —  c  =  0,  ij/ 

will  designate  the  trace   CD,   common  to  the  two  planes. 
The   equation  may  be  placed  under  the  form, 

25  =   —  ax  -{-  c, 

and  hence,   the  trace  may  be   dra^vn.     Or,  if  we  make,  in 
succession, 

X  =  Oy  and  s  =  0, 

we  shall  find. 


=  c 


AD. 


and 


=  AO, 


and  the  trace  may  then  be   drawn  through  the  points   C 
and  D, 

1.    We  likewise  find,  for  the  trace  j57>, 

z  =  ~-  by  +  c; 

a         c 
and  for  the  trace  BC,    y=— ^-aj-l-r-* 

0  0 


We    also    find    AD  =  c,    by  making    y  =  0,    in   the 


190 


ANALYTICAL      GEOMETRY.  [bOOK   VI. 


equation  of  the    trace   BD^ 

and    AB  =   -,    by   making 

jc  =  0,  in  the  equation  of 
the  trace  BC^  or  by  making 
2  =  0,  in  the  equation  of 
the  trace  BB. 

2.  By  comparing  the  equa-  „^ 
tions  of  the  traces  with  the 

equation  of  a  straight  line,  in  Bk.  1.,  Art.  13,  we  see  that, 

—  o,   is  the  tangent  of  the  angle  Tviiich  the  trace   CD 

makes  with  the  axis  of  X; 

—  b,   the  tangent   of  the  angle  which  the  trace  BB 

makes  with  the  axis  of  Y;    and 

—  7 ,  the  tangent  of  the   angle  which  the  trace  B  C 

makes  with  the  axis  of  X. 

3.  The   equation   of  a  plane  may  be  written  under  the 
form, 


Ax  +  Bj/  -{'  Cz  -\-  B  =  0, 


in  which  A,  J5,  (7,  and  B,  are  constant  for  the  same  plane, 
but  have  different  values  when  the  equation  represents  dif- 
ferent planes.  The  coefficients  A,  B,  and  (7,  are  arbitrary 
functions  of  the  angles  which  the  traces  of  the  plane  form 
with  the  co-ordinate  axes,  and  Z>  is  an  arbitrary  function 
of  the  distances  from  the  origin  to  the  points  in  which  the 
plane  cuts  the  co-ordinate  axes.  If  the  plane  passes  through 
the  origin  of  co-ordinates,  its  equation  takes  the  form. 

Ax -{-  Bi/  -h  Oz  =  0. 


BOOK  VI.] 


THE      PLAIJE 


191 


Ziine  perpendicular  to   a  plane. 
27.    The   equations    of  the   straight    line,  to  which   the 
plane  has  been  drawn  perpendicular,  are, 

JC  -  a'  =    a(2  —  Z'),  y  -  y'   -    l){z-  Z')  J 

and  the  equations  of  the  traces 
(7Z>,  j5Z>,  may  be  placed  un- 
der the  form, 

1        c  \        c 

X  =z 2  +  -,   2/  =  —  t2  +  7  • 

a       a  0        0 

By  comparing  the  coeffi- 
cient of  3,  in  the  equation  of 
the  projection  of  the  line  on  ^^ 
the  co-ordinate  plane  ZX^  with  the  coefficient  of  s  in  the 
equation  of  the  trace  (7Z>,  we  find,  that  their  product  plus 
unity  is  equal  to  0 ;  hence,  the  lines  are  at  right  angles  to 
each  other.  The  same  may  be  shown  for  the  trace  ^2>, 
and  the  projection  of  the  line  on  the  plane  YZ ',  and  also 
for  the  trace  jBC,   and  the   projection  on  the   plane    YX. 

Hence,  this  property,  viz. :  If  a  line  be  perpendicular  to 
a  plane  in  space,  the  projections  of  the  line  will  he  re- 
spectively perpendicular  to  its  traces. 


EXAMPLES. 

1.  Find  the  traces  of  a  plane  whose  equation  is, 

2  —  92/  +  11a;  —  12  =  ,0. 

2.  Find  the  traces   of  a  plane   perpendicular  to  a  line, 
whose  equations  are, 

a;  =  32  +  5,         and         y  =   —  2z  —  4, 


192  ANALYTICAL      GEOMETKY.  [bOOK    VI. 

3.    Find  the  traces  of  a  plane  "whose  equation  is, 
2x  —  3y  —  z  =  0. 

SURFACES     OF     THE     SECOND      OEDER. 

28.  The  EQUATION  of  a  sukface,  is  an  equation  express* 
ing  the  relation  between  the  co-ordinates  of  every  point 
of  the  surface. 

It  has  been  shown  (Bk.  I.,  Art.  18),  that  every  equa- 
tion of  the  first  degree,  between  two  variables,  represents 
a  straight  line  ;  and  in  (Bk.  V.,  Art.  40  —  6),  that  every 
equation  of  the  second  degree,  between  two  variables, 
represents  a  circle,  an  ellipse,  a  parabola,  or  an  hyperbola. 

It  has  also  been  shown  (Art.  26 — 3),  that  an  equation 
of  the  first  degree  between  three  variables  represents  a 
plane,  and  analogy  would  lead  us  to  infer  what  may  be 
rigorously  proved,  viz.:  that  everi/  equation  of  the  second 
degree,  between  three  variables,  represe7its  a  curved  surface. 

29.  Surfaces,  like  lines,  are  classed  according  to  the 
degree  of  their  equations.  The  plane,  whose  equation  is 
of  the  first  degree,  is  a  surface  of  the  first  order ;  and 
every  surface  whose  equation  is  of  the  second  degree,  is 
a  surface  of  the   second  order, 

30.  The  equation  of  a  surface,  is  an  equation  which  ex- 
presses the  relation  between  the  co-ordinates  of  every  point 
of  the  surface.  Although  the  equation  determines  the  sur- 
face, yet  it  does  not  readily  present  to  the  mind,  its  form, 
its  dimensions,  and  its  limits.  To  enable  us  to  conceive 
of  these,  we  intersect  the  surface  by  a  system  of  planes, 
parallel  to   the   co-ordinate   planes.      If  then,  we    combin 


BOOK  VI.] 


THE      PLANE, 


193 


the  equations  of  these  planes  with  the  equation  ot  the  sur- 
face, the  resulting  equations  will  represent  the  curves  in 
which  the  planes  intersect  the  surface.  These  curves 
will  indicate  the  form,  the  dimensions,  and  the  limits  of  the 
surface. 

31.     To  give  a  single  example,  let  us  take  the  equation, 

iC2   +    2/2  ^   22     _     722, 

Let  us  intersect  the  surface  re- 
presented by  this  equation,  by  a 
plane  parallel  to  ZX,  and  at  a 
distance  from  it  equal  to  c.  The 
equation  of  the  plane  will  be 
(Art.  11), 

z  =    ±  c. 

Combining  this  with  the  equa- 
tion  of  the   surface,   we   shall  have, 

jc2  +  2/2    ^   7^2  _  ^2^ 

which  is  the  equation  of  the  curve  of  intersection.  This^ 
equation  represents  the  circumference  of  a  circle,  whoso 
centre  is  in  the  axis  of  Z,  and  radius,  ^H^  —  c'^.  The  ra- 
dius will  be  real,  for  all  values  of  c  less  than  J2,  whether- 
c  be  plus  or  mmus.  It  is  zero,  when  c  is  equal  to  i?,  and' 
imaginary,  when  c  is  greater  than  JR.  Tlius,  in  the  first 
case,  the  intersection  will  be  the  circumference  of  a  circle,, 
in  the  second  case,  it  will  be  a  point,  and  in  the  third,  it 
will  be  an  imaguiary  curve ;  or,  in  other  words,  the  plane 
will  not  intersect  the  surface. 

Since    the    given    equation   is    symmetrical    with     respect 
13 


194:  ANALYTICAL      GEOMETRY.  [bOOK  VI. 

to  the  tliree  variables  x,  y,  and  25,  we  may  obtain  similar 
results  by  intersecting  the  surface  by  planes  parallel  to  the 
co-ordinate  planes,  YZ  and  ZJT. 

The  co-ordinate  planes  intersect  the  surface  in  circlep, 
whose   equations   are, 

(C2  +    y2    =    Ji2^  X^  +   Z^    =    I^,  ^^  _|_   ^2    _    J^^ 

These  results  indicate  that  the  surface  whose  equation  is, 

JC2  _|_  2/2  ^_  22    _    7^2^ 

is  the  surface  of  a  sphere  ;  but,  to  prove  it  rigorously,  it 
would  be  necessary  to  show,  that  every  secant  plane  would 
intersect  it   in   the   circumference   of  a  circle. 

Surfaces  of  revolution. 

32.  Every  surface  which  can  be  generated  by  the  revo- 
lution of  a  line  about  a  fixed  axis,  is  called  a  surface  of  revo- 
lution. 

The  revolving  line  is  called  the  generatrix  j  the  line  about 
which  it  revolves,  is  called  the  axis  of  the  surface^  or  the 
axis  of  revolution. 

In  all  the  cases  considered,  we  shall  suppose  the  generatrix, 
in  its  first  position,  to  be  in  the  co-ordinate  plane  ZX,  and 
to  be  revolved  about  the  axis  of  Z. 

33.  When  the  generatrix  is  a  straight  line,  and  not  per- 
pendicular to  the  axis  of  Z,  the  surface  described  is  called  a 
surface  of  single  curvature.  When  the  generatrix  is  a  curv^c, 
the  surface  is  called  a  surface  of  double  c^'rrature. 

The  section  made  by  a  plane  passing  through  the  axis,  ia 
called  a  meridian  section;  or  a  meridian  curve,  when  the 
surface  is  of  double  curvature. 


BOOK  VI.]         SURFACES      OP      REVOLUTION.  195 

34.  It  is  plain,  from  the  definition  of  a  surface  of  revo- 
lution, that  every  point  of  the  generatrix  will  describe  the 
circumference  of  a  circle,  the  centre  of  which  is  in  the  axis. 

Equations  of  the  surfacen. 

35.  Let  DC  he  any  curve,  in 
the  co-ordinate  plane  Z^,  and 
let  it  be  revolved  around  the  axis 
of  ^;  it  is  required  to  determine 
the  equation  of  the  surface  which 
it  will  describe. 

If  we  designate  the  abscissa  of 
any  point  of  the  generatrix,  as  J),  by  r,  and  the  ordinate 
by  z,   the   equation   of  the    generatrix  will  be   expressed   in 
terms  of  r  and  z ;  and  may  be  written  under  the  form, 

r  =  n^); (1.) 

which  is  read,  r,  a  function  of  2,  and  means,  that  r  may  be 
expressed  in  terms  of  z,  and  when  so  expressed,  the  equa- 
tion of  the  generatrix  is  known. 

We  have  now  to  express,  analytically,  the  conditions  which 
will  cause  this  point  of  the  generatrix  to  describe  the  circum- 
ference of  a  circle  around  the  axis  of  Z.  To  do  this,  we 
have  only  to  consider,  that  the  circumference  described  by 
the  point  D,  will  be  projected,  on  the  co-ordinate  plane  YJT, 
into  an  equal  circumference.  If  the  co-ordinates  of  the 
points  of  this  circumference  be  designated  by  x  and  y,  we 
shall  have, 

r  =   ya^jTp (2.) 

If  we  now  suppose  r  to  take  all  possible  values  in  equa- 
tion   (1),    that    will   satisfy  the  equation   of  the   generatrix. 


196  ANALYTICAL      GEOMETRY.  [bOOK   VI. 

and  then  combine  this  equation  with  Equation  ( 2 ),  we  shall 
have, 

r  =  F{2)  =  v^M^^     .    .    .     (3.) 

which  is  the  equation  of  the  surface  of  revolution. 

An  examination  of  the  construction  of  Equation  ( 3 ),  will 
indicate  the  method  of  applying  it,  in  finding  the  equation 
of  any  surface. 

Equation  ( 1 )  is  the  equation  of  the  generatrix  from  which 
r  is  found,  in  terms  of  z  and  constants.  This  confines  the 
point  D,  whose  co-ordinates  are  r  and  2,  to  the  generatrix. 
Equation  (2)  requires,  that  the  curve  traced  by  Z>,  be  the 
circumference  of  a  circle;  hence,  the  combination  of  those 
two  equations,  gives  the  equation  of  the  surface. 

36.  As  a  first  application,  let  it  be  required  to  find  the 
surface  generated  by  the  semi-circumference  of  a  circle, 
whose  centre  is  at  the  origin  of  co-ordinates. 

The  equation  of  the  generatrix  will  be, 

r^+  z"  =  JR^l 

hence,  r  ^  -/i^^  _  g2^ 

•   Substituting  this  value  of  r,  in  Equation  (3),  we  have, 

V-^'  -  22  =   V'a;2  +  2/2  . 

or,  x^  +  2/2  +  s2  _  j^2^ 

which  is  the   equation   of  the  surface   of  the  sphere,  when 
the   centre  is  at  the   origin   of  co-ordinates. 

37.  The  volume  described  by  the  revolution  of  an  ellipse 
about  either  axis,  is  called,  an  ellipsoid  of  revolution.     It 


BOOK   VI.]  SURFACES      OF      REVOLUTION.  197 

is  also,  sometimes,  called  a  spheroid.  It  is  called  a  prolate 
spheroid^  when  the  ellipse  is  revolved  about  its  transverse 
axis,  and  an  oblate  spheroid^  when  it  is  revolved  about  the 
conjugate  axis.  | 

38.  Let  it  be  required  to  find  the  equation  of  the 
surface  of  a  prolate  spheroid.  Since  the  transverse  axis  of 
the  ellipse  coincides  with  the  axis  of  Z,  the  equation  of  the 
generatrix  will  be, 

jBV  ^  ^2^2   ^    ^2^2  . 


,  /32J52  _    ^2^2 

hence,  r  =  ^ -^^ 

Substituting    this  value,  in    the  general  equation  of  the 
surface   of  revolution,  Equation  ( 3 ),  we  obtain, 

which  is  the  equation  of  the  surface  of  a  prolate  spheroid. 

39.  We  find,  by  a  similar  process,  the  equation  of  the 
surface  of  the   oblate  spheroid,   to  be, 

^V  ^   J?2(a;2  +    2/2)     _     ^2J52^ 

If  in  either  of  these  equations,  we  make  -4  =  ^,  we 
obtain, 

a;2+   y2_{.   g2   _    J22, 

the  equation  of  the  surface  of  a  sphere. 

40.  If  an  hyperbola  be  revolved  about  its  transverse 
axis,  each  branch  will  describe  a  volume.  The  surface  of 
each  volume  is  called  a  nappe^  and  the  twp  volumes  taken 


198 


ANALYTICAL      GEOMETET, 


[book  VL 


together,   are   called,   an    hyperboloid  of  revolution  of  two 
nappes. 

The  volume  described  by  the  revolution  of  an  hyj^erbola 
about  its  conjugate  axis,  is  called,  an  hypei'holoid  of  re- 
volution  of  one  nappe. 

41.  If  the  transverse  axis  of  an  hyperbola  coincides 
with  the  axis  of  Z,  the  centre  being  at  the  origin,  ita 
equation   will  be, 

If  the   conjugate   axis   coincides  with   Z,  we  have, 

^2^2   _  ^2^2   _   _  ^2^2^ 

In  the  first   case,   the   equation   of  the  surface  is, 

^2^2  _  ^2(^:2  +  2/2)  —  ^2^2  .   and  in  the  second, 

^2^2   _  ^2(^2   +  2/2)  =   -  A'^B^. 

42.  If  a  parabola  be  revolved  around  its  axis,  the 
volume  described,  is  called  a  paraboloid  of  revolution. 
The   equation  of  the  generatrix  being, 

the   equation  of  the  surface  will  be, 
aj2  ^  2/2  _  2^. 

Surfaces    of   single    curvature. 

43.  Let  the  generatrix  be  a 
straight  line  parallel  to  the  axis 
of  Z.  Equation  (1)  will  then 
become, 

r = l\z)  z=  cf ,  an  arbitrary  constant ;  y 

Vhat  is,  for  every  value  of  z,   r  will  be   constant. 


/^ 


BOOK    VI.]  SURFACE      OP      THE      CONK.  199 

Equation    (2)   will  become, 

r  =   -y/x^  +  y\ 
or,  a;2  _|_  y2  __  ^2 . 

lieuce,  the  equations  of  the   surface,  are, 

r  =  an  arbitrary  constant,      and     tP-  -{-  y'^  —  r^. 

The  first  condition  indicates,  that  every  point  of  tlu; 
generatrix  is  at  the  same  distance  from  the  axis  of  Z; 
hence,  each  point  will  describe  an  equal  circumference. 
The  second  condition  indicates,  that  all  these  circumferences 
will  be  projected  into  the  same  circumference,  on  the  co- 
ordinate plane  YX\  hence,  the  surface  is  that  of  a  right 
cylinder  with  a  circular  base. 

Surface  of  the  cone. 

44.  Let  the  generatrix  be 
any  straight  line  oblique  to  the 
axis  of  Z,  as  BC.  Denote  the 
distance  A  (7,  by  c.  Tlien,  since 
BC  passes  through  the  point 
(j\  whose  co-ordinates  are, 
z'  —  c,  and  x'  —  0,  its  equa- 
ticm   (Bk.  I.,  Art.  20),  is, 

*  z  —  ax  +  c; 

if  r  denote  the  abscissa  of  any  point  of  the  generatrix,  and 
2  the  ordinate,  its  equation  (Art.  35)  will  be, 


ar  -f- 


whence. 


r  = 


200  ANALYTICAL      GEOMETRY 

whence,   vre  have   (Ait.  35), 


[book  Vl 


z—  c 


V^M-y'; 


or. 


(z  —  cy  =  a^ix^  +  2/^; 


(1.) 


In   this   equation,   a  is  the  tangent   of  the   angle    CBX. 
Denote   its  supplement,    CBA^   by  v ;  then,* 


or. 


a^  =  tan^y 


a  =    —  tan  v ; 
hence,   Equation   (l')   becomes, 

(a;2  +  y2)  tan^y  =   (2  —  c)2     ...     (2.) 

This  is  the  equation  of  the  surface  of  the  cone  generated 
by  the  line  B  (7,  revolving  about 
the  axis  Z.  It  is  a  right  cone^ 
with  a  circular  base.  C  is  the 
vertex  of  the  cone,  AB^  the 
radius  of  the  circle  in  which  it 
is  intersected  by  the  co-ordi- 
nate plane  yiY",  and  v,  the  angle 
which  the  generatrix  makes  with 
the  base. 

If    the    generatnx,    BC,    be 
prolonged   beyond  the  point  C, 

the  prolongation  will  generate  an  equal  conical  surface, 
lying  above  the  vertex  C.  The  conical  surface  below  the 
point  (7,  is  called  the  loicer  nappe  of  the  cone,  and  the 
surface   above    (7,   the   upper  nappe   of  the   cone. 

*  Legendre,  Trig.  Art.  63. 


BOOK   YI.]  SUKFACE      OP      THE      CONE. 


201 


45.  Let  the  surface  of  this  cone  be  now  intersected  by 
a  piano,  passing  through  the  axis 
of  Y,  and  consequently  perpen- 
dicular to  the  co-ordinate  plane 
ZJC.  Designate  by  it,  the  angle 
DAJC,  which  the  secant  plane 
makes  with  the  co-ordinate  plane 
yX.  The  equation  of  this 
plane  will  be  the  same  as  that 
of  its  trace  AD  (Art.  18—1); 
that  is, 

z  =  X  tan  u. 

If  we  combine  this  equation  with  the  equation  of  the 
sui-face,  we  shall  obtain  the  equation  of  their  curv^e  of 
intersection.  This  equation  is  of  the  simplest  form,  when 
the  curve  is  referred  to  two  axes  in  its  own  plane.  Let 
us,  then,  refer  it  to  the  two  axes,  A  Y^  AD^  in  the  plane 
of  the   curve,   and   at   right   angles  to   each   other. 

If  we  designate  the  co-ordinates  of  any  point,  referred 
to  these  axes,  the  one,  for  example,  which  is  projected 
at   -5,   by  cc',  y\  we   shall   have, 

AC  =^  X  =  a' cos  I/,        BC  —  z  =  x'sinu; 

and   since  the   axis   of   Y  is  not   changed, 

y  =  y'- 

Substituting  these  values  in  equation  of  the  surface,  (Equa- 
tion  2),  we  shall  obtain,  after  reduction,  the  equation  of 
intersection, 

y'2  tan^y  -f  x"^  co8-i«  (tan-?;  —  tan^w)  -f  lex'  sin  u  =  c^; 
9* 


202  ANALYTICAL      GEOMETRY.  [bOOK    VI. 

or,   omitting   the   accents, 

y2  tan^y  -f-  x"^  cos'^u  (tan^y  —  tan^w)  +  2caj  sin  u  —  c^. 

This  equation  is  of  the  same  form  as  Equation  (4), 
Bk.  v.,  Art.  38  ;  hence,  what  was  proved  of  that,  may  be 
proved  of  this.     Therefore, 

1st.  When  the  coefficients  of  y^  and  a^,  have  the  same 
sign,   the   curve  will  be   an   ellipse  : 

2d.  When  the  coefficient  of  jc^,  is  zero,  the  curve  will 
be  a  parabola  : 

3d.  When  the  coefficients  of  y"^  and  jc^,  have  unlike 
sfigns,   the   curve  will  be   an   hyperbola. 

Since  the  tan^y  is  always  positive,  the  change  of  signs 
in  the  coefficients  of  y'^  and  x\  must  arise  from  the 
change  of  sign  in  the  coefficient  of  x^\  and  since  coshi 
is  positive,  the  sign  of  this  coefficient  wdll  depend  on  the 
relative  values  of  v  and  u.  When  v  >  w,  it  will  be  posi- 
tive ;  when  v  =  u^  it  will  be  zero ;  when  w  >  ^^,  it  will 
be  negative. 

46.  In  order  to  obtain  the  forms  and  classes  of  the 
curves  which  result  from  the  intersection  of  the  cone  and 
plane,  it  might,  at  first,  seem  necessary  to  cause  the  angle 
u  to  vary  from  0  to  3G0°.  But  since  the  sui*iace  of  the 
cone  is  s}Tnmetrical  with  respect  to  its  axis,  it  is  plain 
that  all  the  varieties  will  be  obtained  by  varying  u  from 
0   to   90°. 

47,  Let  us  then   resume  the    equation   of  intersection, 
2/2  tan^y  +  x^  cos^w  (tan^y  —  tan^w)  +  2caj  sin  w  =  c-, 


BOOK  VI.]  SURFACE      OF      THE      CONK. 

and  begin  the  discussion   of  it,  by  supposing, 

u  =  0, 

which  will  cause  the  secant  plane 
to  coincide  with  the  co-ordinate 
plane  ICT.  The  equation  of  the 
curve  will  then  become, 


208 


tan^y ' 


hence,  the  curve  is  the  circum- 
ference of  a  circle,  of  which  A 
IS  the  centre,  and  AD  equal  to 
c 


tany 


,   the  radius. 


48.  If  we  now  suppose  u  to  increase,  the  curve  of 
intersection  will   be   an  ellipse,  so   long  as  u  <C  v;   that  is, 

^  a  right  cone  with  a  circular  base  be  intersected  bij 
a  plane^  maJcing  with  the  base  of  the  cone  an  angle  less 
than  the  angle  formed  by  the  generatrix  and  base^  all  the 
elements  of  the  same  nappe  will  be  intersected.,  and  the 
cij>rve  of  intersection  loill  be  an  ellipse. 


49.  When  u  becomes  equal 
to  V,  the  cutting  plane  becomes 
parallel  to  a  generatrix  of  the 
cone:  hence. 


If  a  right  cone  with  a  circular 
base  be  intersected  by  a  plane 


204 


ANALYTICAL      GEOMETRY.  [bOOK   VI. 


parallel    to    the    generatrix^   the   curve  of  intersection  wiU 
be  a  parabola. 

♦50.  When  u  becomes  greater 
flian  V,  the  cutting  plane  will  in- 
tersect both  nappes  of  the  cone ; 
hence, 

If  a  right  cone  with  a  circular 
base  be  intersected  by  a  plane 
making  with  the  base  of  the 
cone  an  angle  greater  than  the 
angle  formed  by  the  generatrix 
and   base,    both    nappes    of  the 

cone  will  be  intersected,  and  the  curves  of  intersection  are 
hyperbolas. 


DIFFERENTIAL 


AND 


INTEGRAL    CALCULUS, 


DESIGNED      FOR 


ELEMENTARY  INSTRUCTION. 


By    CHAELES    DAVIES,    LL.D., 


PB0FEB80K  OF  HIGHEB  MATHEMATICS,  COLrMBIA  OOLLEQB. 


Emtbbei),  according  to  Act  of  Congress,  in  the  year  Eighteen  Hundred  and  Sixty. 

Bt     CHAELES     DAVIE? 

In  the  Clerk's  Office  of  the  District  Court  for  the  Southern  District  of  New  Tork, 


PREFACE. 


The  Diffeeentiai,  aitd  Integrai.  Calculus  is  too  impor- 
tant a  part  of  Mathematical  Science  to  be  entirely  omitted 
in  a  course  of  Collegiate  instruction. 

The  abstract  quantities,  Number  and  Space,  are  pre- 
sented to  the  mind,  in  the  elementary  branches  of  Mathe- 
matics, as  of  definite  extent,  and  as  made  up  of  parts; 
and  the  value  or  measure  (how  much?)  in  any  given  case, 
is  expressed  by  the  number  of  times  which  the  quantity 
contains  one  of  its  parts,  regarded  as  a  unit  of  measure. 
But  we  do  not  attain  to  a  clear  apprehension  of  their 
quantitative  nature^  until  we  regard  them  as  of  indefinite 
extent,  as  possessing  continuity,  and  as  capable  of  chang- 
ing from  one  state  of  value  to  another,  according  to  any 
conceivable  law. 

The  Difierential  and  Integral  Calculus  embraces  all  the 
processes  necessary  to  such  an  analysis.  It  regards  quan- 
tity as  the  result  of  change.  It  examines  established  laws 
of  change  and    determines  their  consequences.     It  supposes 


IV  •  PREFACE. 

laws  of  change  and  traces  the  results  of  the  hypothesis. 
In  short,  it  embraces  -within  its  gras}D — in  the  Material, 
everything  from  the  minuest  atom  to  the  largest  body — ^in 
Space,  all  that  can  be  measured,  from  the  geometrical  point 
to  absolute  infinity — in  Time,  the  entire  range  of  duration 
— and  in  Motion,  every  change  from  absolute  rest  to  infinite 
velocity. 

The  German,  the  French,  the  English,  and  even  the 
American  press,  has  been  prolific  in  the  number  of  Trea- 
tises recently  published  on  this  subject.  The  effort  to 
furnish  better  Text-Books,  proves  at  once  the  value  of  the 
knowledge,  and  the  great  difficulty  of  presenting  it  in  the 
best  possible  form. 

In  regard  to  the  Treatise  now  presented  to  the  public, 
I  have  simply  to  say,  that  it  is  an  Elementary  Text-Book 
for  the  use  of  College  Classes,  and  other  classes  of  about 
the  same  grade.  The  Treatise  of  Professor  Courtexay, 
late  of  the  Virginia  University,  and  that  of  Professor 
Church,  of  the  Military  Academy,  may  be  advantageously 
read  by  those  who  wish  to  advance  further;  and  it  is 
due  to  Columbia  College  to  state  that  this  Treatise  is  used 
in  the  Course  prescribed  to  all  the  pupils,  and  is  not  an 
exponent  of  the  higher  course  pursued  by  those  who  make 
Mathematical  Science  a  special  branch. 

Columbia  Collegb,  June,  1860. 


CONTENTS. 


SECTION    I. 

DEFINITIONS      AXi)      FIS  ST      PRTKCIPLE8. 

UtTIOLS 

Dcfinitiona 1 

Uniform  and  Varying  Changes 2-4 

Function  and  Variable 4-9 

Algebraic  and  Transcendental  Functions 9 

Geometrical  representation  of  Functions 10 

Language  of  Numbers  inadequate 11 

Consecutive  Values  and  DifiFerentials. ...    12 

Differential  Coefficient 13 

Form  of  difference  between  two  states  of  a  Function 14-15 

Differential  Coefficient  and  Differential 15 

Equal  Functions  have  equal  Differentials 16 

Converse  not  True 17-1 0 

Signs  of  the  Differential  Coefficient 19 

Nature  of  a  Differential  Coefficient,  and  of  a  Differential 20 

Rate  of  Change 21-24^ 

Nature  of  Diflferential  Calculus 24-26 . 


SECTION  II. 

D IFr EBE NTI AL3      OF      ALGEBRAIC      KUNOTIONa. 

Differential  of  Sum  or  Difference  of  Functions 26 

Differential  of  a  Product    27-29 

Differentials  of  Fractions 20 

Differentials  of  Powers  and  Formulas 30 

Differential  of  a  Particular  Binomial 30 

Riite  of  Clian-e  of  a  Function 31  • 

Partial  DiiTcrentiaLs 32-.-34 

14 


VI  CONTENTS 


SECTION  IIL 


INTEGRATION      AND      APPLICATIONB. 

▲BTIOLa 

Integration  and  A  pplications 34 

Integration  of  Monomials 35-41 

Integration  of  Particular  Binomials 41 

Integration  by  Series   42 

Equations  of  Tangents  and  Normals 43-50 

Asymptotes 60-52 

Differential  t  f  an  Arc 52 

Rectirteation  of  Plane  Curves 53 

Quadratures   54 

Quadrature  of  Plane  Figures 55 

Nature  of  the  Integral 56 

Area  of  a  Rectangle 57 

Area  of  a  Triangle 58 

Area  of  a  Parabola 69 

Area  of  a  Circle 60 

Area  of  an  Ellipse 61 

Quadrature  of  J^urfaces  of  Revolution 62 

Surface  of  a  Cylinder 63 

Surfiice  of  a  Cone 64 

Surface  of  a  Sphere 65 

Surface  of  a  Paraboloid 66 

Surface  of  an  Ellipsoid 67 

Cubature  of  Volumes  of  Revolution 68 

Examples  in  Cubature 68-71 


SECTION  rv. 

BT70CE89IVK      D  T  F  F  E  K  E  X  T  I  A  L  S — SIGNS      OF     DIFFERENTIAL      CO- 
EFFICIENTS— FORMULAS      OF      DEVELOPMENT. 

Succepsive  Differentials 71 

Pigns  of  the  First  Differential  Coefficient 72 

Si'jns  of  the  Second  Differential  Coefficient 73 

Applications   74 

Maclaurin's  Theorem 75-76 

Taylor's  Theorem 77-81 


CONTENTS.  Vii 


SECTION  V. 

MAXIMA      AND      MINIMA. 

ABTrCLP 

Maxima  and  Minima 81-86 

Points  of  Inflection 84 


SECTION  VI. 

DIFFEKENTIALS      OF     TRANSCENDENTAL     FUNCTIONS. 

Differentials  of  Logarithmic  Functions 86 

Relation  between  a  and  k 87-90 

Differential  Forms  which  have  Known  Logarithmic  Integrals 91 

Circular  Functions 92-99 

Differential  Forms  which  have  Known  Circular  Integrals 99 

Applications 100 


SECTION  vn. 

TBANSCENDENT  AL     CURVES — CURVATURE — BADIU8     OF     CURVA- 
TURE—  INVOLUTES     AND      EV0LUTE3. 

Claa8i6cation  of  Curves 101 

Logarithmic  Curve — General  Properties 102-106 

Asymptote 106 

Sub-tangent 107 

The  Cycloid 108 

Transcendental  Equation  of  the  Cycloid 109 

Differential  Equation 110 

Sub-tangent — Tangent — Sub-normal — Normal Ill 

Position  of  Tangent 112 

Curve  Concave 113 

Area  of  the  Cycloid 114 

Area  of  Surface  generated  by  Cycloidal  Arc 116 

Volume  generated  by  Cycloid 116 

Spirals,  or  Polar  Lines 117 

General  Properties 118 

Spiral  of  Archimedes 119 

Parabolic  Spiral 120 

Hyperbolic  Spiral 121 


Viii  CONTENTS. 

▲XTICU 

Logarithmic  Spiral 122 

DirectioD  of  the  Measuring  Arc 123 

Sub-tangent  in  Polar  Curves 124-127 

Angle  of  Tangent  and  Radius-vector 127 

Value  of  the  Tangent 128 

Diflferential  of  the  Arc 129 

Differential  of  the  Area 130 

Areas  of  Spirals v 131-135 

CURVATURE. 

Curvature  of  a  Circle  inversely  as  the  Radius  135-136 

Orders  of  Contact 137 

Oscillatory  Curves 138 

Osculatory  Circle 139 

Limit  of  the  Orders  of  Contact 140 

Radius  of  Curvature 141 

Measure  of  Curvature 142-144 

Radius  of  Curvature  for  Lines  of  the  Second  Order 144-149 

Evolute  Curves .< 149 

A  Normal  to  the  Involute  is  Tangent  to  the  Evolute 150 

Evolute  and  Radius  change  by  Same  Quantity 151 

Evolute  of  the  Cycloid '. 152 

Equation  of  the  Evolute  Curve 153 

Evolute  of  the  Common  Parabola 154 


INTEGRAL  CALCULUS. 

Nature  of  Integration 1 55 

Forms  of  Differentials  having  known  Algebraic  Functions 156-159 

Forms  of  Differentials  having  known  Logarithmic  Functions 159 

Forms  of  Diflferentials  having  known  Circular  Functions 1 60 

Integration  of  Rational  Fractions •  161 

Integration  by  Parts 162 

Integration  of  Binomial  Differentials 163 

When  a  Binomial  can  be  Integrated 164 

Formula    ^ 165 

Formula    ^ 166 

Formula   <5 167 

Formula   32> 168 

Formula   ^ 169 


INTRODUCTION. 


Common  Teems  must  always  be  employed  in  definitions, 
because  a  definition  refers  to  a  class  of  things  in  which 
each  enjoys  at  least  one  property  common  to  all  the 
others.  Each  individual  of  a  class,  so  defined,  is  called  a 
significate.  A  common  term  does  not  express  to  the  mind 
a  distinct  and  adequate  idea  of  any  one  of  its  significates, 
but  a  general  notion  of  them  all ;  hence,  we  do  not  com- 
prehend the  full  scope  and  meaning  of  a  definition,  until 
we  have  ascertained,  by  careful  analysis,  the  number,  of  its 
significates  and  the  exact  characteristics  of  each. 

Mathematics  is  the  science  which  treats,  primarily^  of 
the  relations  and  measures  of  quantities;  and  secondarily, 
of  the  operations  and  processes  by  which  these  relations 
and  measures  are   ascertained. 

QuANTriY  is  anything  that  can  be  increased,  diminished, 
and  measured.  There  are  two  general  kinds.  Number  and 
Space,  and  each  is  subdivided  into  four  classes.  Under 
Number,  w^e  have  Abstract  Number,  Currency,  Weight, 
and  Time;  and  under  Space,  Length,  Surface,  Volume, 
and  Angular  Measure. 

Mathematics,  considered  as  a  science  of  exact  relation, 
is  divided  into  three  branches;  1.  Arithmetic.  2.  Geome- 
try.    3.  Analysis. 


X  INTRODUCTION. 

AKirnMETic  is  that  branch  which  treats  of  the  properties 
and  relations  of  Numbers,  when  expressed  by  figures. 

Geoaietry  treats  of  the  properties  and  relations  of  Mag- 
nitudes, by  reasoning  directly  on  the  magnitudes  them- 
selves, or  upon  their  pictorial  representations.  The  magni- 
tudes considered  in  this  branch  of  Mathematics,  are,  lines, 
surfaces,  volumes,  and  angles. 

Analysis  embraces  all  that  portion  of  Mathematics  in 
which  the  quantities  considered  are  represented  by  letters, 
and  the  operations  to  be  performed  are  indicated  by  signs, 
or  conventional  symbols.  Its  elementary  branches  are,  Al- 
gebra, Analytical  Geometry,  and  Analytical  Trigonometry. 
In  these  branches,  quantities  of  the  same  kind  are  compared 
by  means  of  their  unit  of  measure,  which  has  a  fixed  value, 
and.  is  generally  expressed,  numerically,  by  the  unit   1. 

A  Variable  Quantity  is  one  which  increases  or  dimi- 
nishes, according  to  any  law,  and  thus  passes  from  one 
state  of  value  to  another.  If,  in  changing  its  value,  be- 
tween any  two  limits,  it  passes  through  all  the  interme- 
diate values,  it  is  called  a  continiioxis  quantity. 

Two  VALUES  OF  A  VARIABLE  QUANTITY  are  consccutive, 
when  the  greater  cannot  be  diminished,  according  to  the 
law  of  change,  without  becoming  equal  to  the  less.  Hence, 
there  is  no  intermediate  value. 

When  we  say  that  a  continuous  quantity  passes  from  one 
state  of  value  to  another,  we  mean  that  it  either  increases 
or  diminishes;  and  when  we  speak  of  the  next  value,  we 
mean  the  first  value  which  it  assumes  when  the  change 
begins.    These  two  values  are  consecutive. 


INTRODUCTION.  XI 

If  we  suppose  a  variable  quantity,  denoted  by  x,  to  have 
a  particular  value,  x  =  a^  and  afterwards  to  assume  another 
value,  X  =  «',  we  may  suppose  that  x  changes  uniformly 
from  a  to  a',  and  assumes,  in  succession,  all  the  values 
between  its  limits.  These  two  suppositions  render  it  impos- 
sible to  express  the  change  in  value,  either  by  1,  or  by  any 
of  the  parts  of  1. 

For,  denote  the  uniform  change  in  x^hy  h:  then,  if  h 
could  be  expressed  by  a  fiaction,  however  small,  that  frac- 
tion could  be  diminished  by  increasing  its  denominator : 
hence,  there  would  be  values  between  a  and  «',  through 
which  x  would  not  pass,  which  is  contrary  to  the  hypoth 
esis.  Therefore,  the  hypothesis  that  x  changes  uniformly, 
and  passes  through  all  values  between  x=^  a  and  a^  =  a', 
renders  it  impossible  to  express  the  change  of  value  by 
numbers.  Therefore,  when  the  change  is  uniform,  the  dif- 
ference between  consecutive  values  cannot  be  expressed  by 
1,  or  in  parts  of  1.  The  same  is  also  true  when  the  change 
is  not  uniform.  For,  if  the  difference  of  two  consecutive 
values  could  be  expressed  by  one,  or  in  parts  of  1,  it  could 
be  diminished ;  hence,  there  would  be  intermediate  values, 
which  is  contrary  to  the  definition. 

The  hypothesis,  therefore,  of  continuous  quantity,  renders 
it  impossible  to  express  the  elementary  changes  of  value  by 
means  of  numbers ;  and  hence,  we  are  unable  to  deal  with 
such  changes  by  any  of  the  methods  of  calculation  already 
explained. 

The  Differential  and  Integral  Calculus  is  the  name 
given  to  that  branch  of  Mathematics  which  treats  of  the 
properties  and  relations  of  continuous  quantities — of  the  laws 


Xn  IS  T  li  O  D  U  C  T  I  O  N . 

of  change  to  wliich  tlicy  may  be    subjected,  and  of  the   re- 
sults flowing  from  sucli  changes. 

When  we  measure  a  quantity,  great  or  small,  the  stand- 
ard or  unit  of  measure,  is  of  the  same  kind  as  the  quantity 
measured,  and  the  ratio  of  this  unit  to  the  quantity,  is  the 
result  of  the  measurement.  In  the  operations  of  the  Cal- 
culus, the  assumed  unit  of  measure  is  the  change  which 
takes  place  in  a  quantity  that  varies  uniformly.  This  quan- 
tity is  called  the  i?idepe?ide?it  variable.  The  quantity  whose 
changes  of  value  are  measured,  is  called  a  function.  The 
independent  variable  and  function  are  connected  by  a  law, 
either  expressed  or  implied,  and  change,  simultaneously,  ac- 
cording to  that  law. 

The  theory  of  the  Calculus,  therefore,  rests  on  the  fol- 
lowing axioms  and  inferences : 

1.  Where  no  law  of  change  has  been  fixed,  such  a  law 
may  be  imposed  as  will  cause  a  variable  to  change  uni- 
formly, and  to  pass  through  all  values  betn-^en  any  two 
limits. 

2.  The  diflerence  between  any  two  consecutive  values  of 
a  quantity  so  varying,  is  constant. 

3.  The  ratio  of  the  smallest  change  in  the  independent 
variable,  to  the  corresponding  change  of  the  function,  de- 
termines the  rate  of  change  of  the  function ;  and  the  actual 
change  is  denoted  by  this  rate  into  the  change  of  the  in- 
dependent variable. 


DIFFERENTIAL    CALCULUS. 


SECTIOlSri. 

DEFINITIONS    AND    FIRST   PRINCIPLES. 
Definitions. 

1.  In  the  Differential  Calculus,  as  well  as  in  Analyt- 
ical Geometry,  the  quantities  considered  are  divided  into 
two  classes : 

■*st.  Consta7it  quantities^  ichich  preserve  the  same  values 
tn   the   same  investigation/   and, 

2d.  Variable  quantities,  which  assume  all  possible  values 
that  will  satisfy  any  equation  ichich  exjpresses  the  relatio?i 
between  them. 

The  constants  are  denoted  by  the  first  letters  of  th« 
alphabet,  «,  5,  c,  &c. ;  and  the  variables,  by  the  final  let- 
ters, jc,  y,  0,  &c. 

Uniform    and   var37ing    changes. 

2,  There  are  two  ways  in  which  a  variable  quantity 
may  pass   from   one   value   to   another. 

If  the  variable  jc,  once  had  the  particular  value,  a;  —  a, 
and  afterwards  assumed  the  value,  x  =  a\  we  can  sup- 
pose : 


14  DIFFERENTIAL      CALCULUS.  [SEC.    L 

1st.  That  during  the  change  from  a  to  a',  x  assumed, 
in  succession,  and  by  a  uniform  change,  all  the  values 
between  a  and  a',  just  as  a  body  moving  uniformly  over 
a  given  straight  line  passes  through  all  the  points  between 
its  extremities;  or, 

2d.  "We  may  suppose,  that  during  the  change  from  a  to 
a\  X  assumed  all  possible  values  between  its  limits,  with- 
out the  condition  of  a  uniform  change.  In  both  cases, 
the   quantity  is  said  to   be   continuous. 

3,  K  two  variable  quantities,  y  and  x,  are  connected 
in   an   equation,   as,  for   example, 

2/   =  ic2  +  2  ; 

then,  to  every  value  of  a,  arbitrarily  assumed,  there  will 
be  a  corresponding  value  of  y,  dependent  upon^  and  result- 
ing from^  the  value  attributed  to  x.  Thus,  if  we  make 
jc  =  4,    we  have, 

2/  =  16  +  2   =   18. 

If  we  suppose  x  to  increase  from  4  to  5,  we  ghall  have, 

y  =   25  -h  2   =   27; 

thus,  while  x  changes  from  4  to  5,  y  changes  from  18  to  27. 
If  now  we  suppose  x  to  increase  from  5  to  6,  y  "\vill 
increase  from  27  to  38.  Thus,  while  jb  increases  unifonnly 
by  1,  y  will  change  its  value  according  to  a  very  different 
law. 

Function  and  variable. 

4.  When  two  variable  quantities,  y  and  a;,  are  con- 
nected in   an   equation,    either  of  them    may  be   supposed 


SEO.    I.]  FIRSTPRINCIPLES.  15 

to  increase  or  decrease  iiiiifornily ;  a  variable,  so  changinsj, 
is  calletl  the  independent  variable^  because  the  law  of 
change  is  arbitrary^  and  independent  of  tlie  form  of  equa- 
tion. This  vaiiable  is  generally  denoted  by  a?,  and  called 
simply,  the  variable.  The  change  in  the  variable  y,  de- 
pends on  the  form  of  the  equation  ;  hence,  y  is  called 
the  dependent  variable,  or  function.  When  such  a  rela- 
tion exists  between  y  and  cc,  it  is  expressed  by  an  equa- 
tion  of   the   form, 

y  =  F{x\        y  =  f{x) ;         or,        /(y,  a;)    =  0  ; 

which  is  read,  y  a  function  of  x.  The  letter  i^  or  y, 
is  a  mere  symbol,  and  stands  for  the  word,  function.  K 
y  is  a  function  of  ic,  that  is,  changes  with  it,  x  is  also 
a  function   of  y;  hence. 

One  quantity  is  a  function  of  another^  when  the  two 
are  so  connected  that  ariy  chojige  of  value.,  in  either.,  pro- 
duces  a  corresponding   change   in   the   other. 

5.  If  the  equation  connecting  y  and  ic,  is  of  such  a 
form  that  y  occurs  cdone^  in  the  first  member,  y  is  called 
an  explicit  function  of  x.       Thus,  in  the  equations, 

y  =  ax  -\-  b of  a  straight  line, 

y  =    'yfl^  —  aj2  .     .     .     .     of  the  circle, 

y  =  -    ^^/A?  —  x^   ,     .     .     of  the  ellipse, 

y  =   ■y/2px of  the  parabola,  and 

B 


y  —   --  ^x^  —  A^   .     .     ,     of  the  hyj  erbola, 
y  is  an   explicit  function   of  x. 


1*6  DIFFERENTIAL      CALCULUS.  [SEC.    I. 

But,   if  the    equations   are    written    under   the   forms, 


y  —  ax  —  h  =   0 ; 

or. 

A^.  y)  =  0, 

if+x^-E^  =  0; 

or. 

f{x,y)   =  0, 

^V  ^  ^2^.2  _  ^2^2  ^   0  ; 

or. 

/(^,  y)  -  0, 

2/2  _  2j:>x  =  0  ; 

or. 

/(aj,  y)  =  0, 

^2y2  _  jj2aj2  ^  ^2^2  3^  0 ; 

or. 

/(^,  y)  =  0, 

y  is  called  an  imjjlicU  function  of  x;  the  nature  of  the 
relation  between  y  and  x  bemg  implied,  but  not  developed 
in  the   equation. 

6.  It  is  plain,  that  in  each  of  the  above  equations 
the  absolute  value  of  y,  for  any  given  value  of  a,  will 
depend  on  the  constants  which  enter  into  the  equation ; 
this  relation  is  expressed,  by  calling  y  an  arbitrary  function 
of  the  constants  on  which  it  depends.  Thus,  in  the  equa- 
tion of  the  straight  line,  y  is  an  arbitrary  function  of  a 
and  b;  in  the  equation  of  the  circle,  y  is  an  arbitrary 
function  of  -R ;  in  the  equation  of  the  ellipse,  of  A  and 
J? ;  in  the  equation  of  the  parabola,  of  2p ;  and  in  the 
equation    of  the   hyperbola,    of  A    and   J). 

7,  An  increasing  function  is  one  which  increases  when 
the  variable  increases,  and  decreases  when  the  variable 
decreases.  A  decreasing  function  is  one  which  decreases 
when  the  variable  increases,  and  increases  when  the  variable 
decreases. 

In  the  equation  of  a  straight  line,  in  which  a  is  posi- 
tive, y  is  an  increasing  function  of  x.  In  the  equations 
of  the  circle  and  ellipse,  y  is  a  decreasing  function  of  x. 
In  the  equation  of  the  parabola,  y  is  an  increasing  func- 
tion of  X.     In  the  equation  of  the  hyperbola,  y  is  imaginary 


SEC.    I.]  FIKST      PKINCIPLES.  17 

for  all  values  of  x  <  A,  and  an  increasing  function  for 
all  2)ositive  values  of  ic  >  ^. 

8,  A  quantity  may  be  a  function  of  two  or  more  vari^ 
ables.      If 

u  =  ax  -{■  hxp-^      or      w  =  aa;^-  —  hy"^  -\-  cz  -\-  d, 

u  will  be  a  function  of  x  and  y,  in  the  first  equation, 
and  of  X,  y,  and  s,  in  the  second.  These  expressions  may 
be   thus  written: 

w  =  /(«,  y),        and        u  =  fix,  y,  2). 

If,  in  the  second  equation,  we  make,  in  succession,  the 
independent  variables  a,  y,  and  2,  respectively  equal  to 
0,   we  have, 

for,  x=0,  u=  ~by^-\-cz+d  =/{'(/,  z), 

for,  JB=0,  and  y  =  0,     u  z=z  cz  +  d  =f('^)  y    and, 

for,   x=0,  y=0,  and  z  =  0,     u  =  d  =a  constant. 

Algebraic  and  Transcendeiital  Functions. 

9.  There   are  two    general    classes   of  functions : 
Algebraic  and    Tra7iscendental. 

Algebraic  functions  are  those  in  which  the  relation  be 
tween  the  function  and  the  variable  can  be  expressed  in 
the  language  of  Algebra  alone :  that  is,  by  addition,  sub- 
traction, multiplication,  division,  the  formation  of  powers 
denoted  by  constant  exponents,  and  the  extraction  of  roots 
indicated  by   constant  indices. 

Transcendental  functions  are  those  in  which  the  relation 
between  the  function  and  variable  cannot  be  expressed  m 
the  language  of  Algebra  alone.       There  are  three  kinds : 


18 


DIFFERENTIAL      CALCULUS. 


[SEa   I. 


1.  Exponential  functions,  in  wliich  the  variable  enters  as 

an  exponent ;   as, 

u  =  a'. 

2.  Logarithmic    functions,  which  involve    the   logarithm 

of  the  variable ;   as, 

u  —  logjc. 

3.  Circular  functions,  which  involve  the  arc  of  a  circle, 
or  some  function   of  the   arc;   as, 

w  =  sin  ic,        w  =  cos  a,        u  =  tan  x. 


Geometrical  representation  of  Functions. 

10.  With  the  aid  of  Anal}i:ical  Geometry,  it  is  easy  to 
trace,  geometrically,  the  numerical  relation  between  any 
function   and  its  independent  variable. 

Suppose  we   have  given  the   equation, 

y  =  fi^)' 

If  we  attribute  to  x,  the  independent  variable,  in  succes- 
sion, every  value  between  —  oo  and  +  oo,  each  will  give 
a  corresponding  value  for  y,  which  may  be  determined 
from  the   equation,    y  =  f{x) 

Let  0  be  the  origin  of  a 
system  of  rectangular  co-or- 
dinates. From  0,  lay  off  to 
the  right,  all  the  positive 
values  of  jc,  and  to  the  left  all 
the  negative  values.  Through 
the  extremity  of  each  abscissa, 
so  determined,  draw  a  line 
parallel  to  the  axis  of  ordinates, 
and  equal  to  the  corresponding  value  of  y;  the  plus  valoef 


T 

y 

0              PC 

\       X 

SEC.   I.] 


FIRST      PRINCIPLES, 


19 


will  fall  above  the  axis  of  JT,  and  the  negative  values  below 
it;  then  trace  a  curve,  AMN^  through  the  extremities  of 
these  ordinates.  The  co-ordinates  of  this  curve  will  indicate 
every  relation  between  y  and  a;,  expressed  by  the  equation, 

y  =  A^)' 
This  curve  should  present  to  the  mind,  not  merely  any 
particular  value   of  x,   and  the   corresponding  value   of  y, 
but  the  entire  series  of  corresponding  values  of  these  two 
variables. 

Quantities  infinitely  small — Differentials. 

11.    A   quantity   is    infinitely   S7nall,   when    it   cannot   be 
diminished,  according  to  the  law 
of  change,  without  becoming  0. 

If,  in  the  equation  of  the  curve, 

y  =fix)  .    .    (1.) 

ic  has  a  particular  value  OP,  y 
will  denote  the  ordinate  PM, 

If  X  be  increased  by  PQ,  de- 
noted by  A,  PiKf  will  change  to 
QN,  which  we  will  denote  by  y' \  and  we  shall  have, 

y'=f{x^-K) (2.) 

If  we  subtract  equation  (1)  from  (2),  we  obtain, 

y'-y=A^  +  K)-f{x)    ....   (3.) 

It  is  evident  that  each  member  of  this  equation  will  reduce 
to  0,  when  we  make  A  =  0. 


T 

0                P         ( 

20 


DIFFERENTIAL      CALCULUS 


Let  US  suppose,  -as  before,  that  the  abscis  x  has  increased  from 
OP  to  OQ,  aud  that  the  cor- 
responding ordinate  y,  has  be- 
come 2/'.  Draw  through  the 
points  iV  and  3/,  the  secant 
line  iV^J/i  If,  now,  we  suppose 
the  point  iV  to  approach  3f, 
till  it  becomes  consecutive  with 
it,  then, 

1.  The   secant   line   will   become  the  tangent  S3IT\ 

2.  The   abscissas   OP   and    OQ  will  become   consecutive; 

3.  The   ordinates  PJf,   QN'^  will  also   be   consecutive. 

The  DIFFERENTIAL  of  a  quantity  is  the  difference  between 
any  two  of  its  co7isecutive  values;  hence,  it  is  indefinitely  small. 

The  differential  is  expressed,  by  writing  d  before  the 
letter  denoting  the  quantity :  thus,  dx  denotes  the  differen- 
tial of  ar,  and  is  read,  differential  of  jc:  dy  denotes  the 
differential  of  y,  and  is  read,  differential  of  y. 

It  is  plain,  that  dx  denotes  the  last  value  of  A,  in  Equa- 
tion (3),  before  it  becomes  0;  and  that  dy  denotes  the  last 
difference  between  y'  aud  y,  as  h  approaches  to  0. 


Differential   Coefficient. 

12.   Under  the   preceding   hypotheses,  the  differentials   of 
X  and  y  admit  of  geometrical  interpretations. 

If  we  divide  both  members  of  Equation  (3)  by  A,  we  have, 


y  -  y 


f{x  +  h)  -f(x) 
h 


(4) 


Having  drawn  MR  parallel  to  the  axis  of  abscissas,  NR 
will  denote  the  difference  of  the  two  ordinates  y'  and  y; 
hence,  the  first,  and  consequently  the  second  member  of 
Equation  (4),  will  denote   the   tangent   of  the   angle  NMR^ 


SEC.    I.]  FIESTPRINCIPLES.  21 

which  the  secant  line  makes  with  the  axis  of  ^.  Denote 
the  angle  which  the  tangent  makes  willi  the  axis  of  JT  by  a. 
When  the  ordinates  y'  and  y  become  consecutiv(\  the 
secant  iO/  becomes  tangent  to  the  curve  at  the  point  J/, 
and  the  angle  NMR  becomes  equal  to  2/S^P;  and  we  have, 

^  =  tana (5.) 

cly 
Tlie  terra   ~-^   is   called   the   differential    coefficient   of  y, 

regarded   as  a  fraction  of  a;;  hence, 

The  differential  coefficient  of  a  fmiotion  is  the  differen- 
tial of  the  function  divided  by  the  differential  of  the 
mdepefide?it  variable. 

If  we   multiply  both   members  of  Equation  (5)  by  dxy 

-^dx  =  tan  a  dx; 
dx 

but  the  tan  a  multiplied  by  the  base  dx,  of  the  indefinitely 
small  triangle,  is  equal  to  the  perpendicular,  which  is  the 
diiference  between  the  consecutive  values  of  y'  and  y,  de- 
noted by  dy;  therefore, 

-^dx  z=  dy ;  hence, 
dx 

The  differential  of  a  function  is  equal  to  its  differential' 
coefficient  multiplied  by  the  differential  of  the  variable. 


Limiting  Ratio. 
13.  Let  us  now  resume  the  consideration  of  Equation  (4). 


h  h 

The    first    member    of  this    equation    is    the   ratio    of  tlie 
increment  A,  of  the  independent  variable,  to  the  correspond 


22  DIFFERENTIAL       CALCULUS.  [SEC.    I. 

ing  increment  of  the  function,  and  denotes  the  tangent  of 
the  angle  wliicli  tlie  secant  line,  drawn  through  the  ex- 
tremities of  y  and  y\  makes  with  the  axis  of  abscissas. 

If  we  suppose  h  to  decrease,  the  secant  line  "will  ap- 
proach the  tangent,  and  the  ratio  will  approach  the  tangent 
of  the  angle  which  the  tangent  line  makes  with  the  axis  of 
X.  The  tan  a,  is,  therefore,  the  limit  of  this  ratio,  and 
since  it   is   also   the  ditferential   coefficient,  it  follows  that. 

The  differential  coefficient  is  the  limit  of  the  ratio  of  the 
increment  of  the  indejyendent  variable  to  the  increment  of 
the  function. 

A  varymg  ratio,  of  any  increment  of  the  independent 
variable  denoted  by  A,  to  the  corresponding  increment  of 
the  function,  denoted  by  y'  —  y,  reaches  its  limit  when  h 
reaches  its  last  value;  and  then,  the  values  of  y'  and  y 
become  consecutive;  therefore,  the  limiting  ratio,  is  the 
ratio  of  consecutive  values.  Hence,  if  we  have  an  expres- 
sion for  the  ratio  of  the  increments,  we  pass  to  the  limiting 
ratio,  or  differential  coefficient,  by  making  h  indefinitely 
small. 

Form  of  the  difference  between  two  states  of  a  function. 
14.    Let  us  resume  the  discussion  of  Equation  (3). 

If  h  be  made  equal  to  0,  the  first  and  second  members 
will  each  reduce  to  0.  Therefore,  if  the  second  member  be 
developed,  and  the  like  terms  having  contrary  signs  can- 
celled, each  of  the  remaining  terms  will  contain  h ;  else,  all 
the  terms  would  not  reduce  to  0,  when  A  =  0.  Hence,  the 
second  member  of  Equation  (3)  is  divisible  by  h.  Dividing 
by  A,  we  have, 

y'-y  _  A^+  f^^  -J'-''  .     .     .     (4.) 


SEC.    I.]  FIRSTPRINCIPLES.  23 

If,  now,  we  pass  to  the  limiting  ratio,  by  making  A  in- 
definitely small,  the  second  member  will  become  the  tan  a, 
a  quantity  independent  of  A  (see  Equation  5);  hence,  the 
first  term  in  the  development  of  the  second  member  of 
Equation  (3)  contains  A  only  in  the  first  power,  and  the 
coefficient  of  this  terra  is  tan  a,  or  the  differential  co- 
efficient. Since  all  the  other  terms  become  0,  when  A  =  0, 
each  of  them  must  contain  A  to  a  higher  power  than  the 
first. 

If  we  designate  by  P,  the  differential  coefficient  of  y,  and 
by  P'  such  a  value  that  P'li^  shall  be  equal  to  all  the 
terms  of  the  development  of  the  second  member  of  Equation 
(3),  after  the  first,  that  equation  may  be  written  under  the 
form 

2/'_y  =  PA  +  P'A'     ....     (6.) 

The  differential  coefficient,  P,  is  independent  of  A,  bat 
will,  in  general,  contain  aj;  and  when  it  does,  it  is  a  func- 
tion of  that  variable ;  P',  when  not  equal  to  0,  is  a  function 
of  X  and  A. 

Applications  of  the  Formula. 
1.    If  we  have  an  expression  of  the  form, 
y  =/W  =  «a;, 
we  have   the   form     or   development    of  the    second   mem- 
ber. 

If  we  give  to  x  an  increment  A,  we  have, 

^y'  =/(a;  +  A)  =     a(a;    +     A)     =z  ax  +aA;  hence, 
V'  -y  =A^  +  h)  -A^)  =  «>^;  and 

y       =  «^ ;  passing  to  consecutive  values, 
♦         ill 

-^  =  a;     and      Jldx  =  adx. 
dx  dx 


24  DIFFEBENTIAL      CALCULUS.  [sEC.   U 

2.    If  we  have  a  function  of  the  form, 
y  =f{^)  =  ax^, 

we    again   have    the   form    or   development    of  the    second 
member. 

y'  =/(x  +  h)  =  a{x  +  hy  z=  ax^  +  Sax^h  +  3axh^  +  ah^ 

y'  -.y=f{x  +  h)  —f{x)  =  Sax^h  +  3axh^  +  ah^ 

y  ^  y  =  3ax2  +  Saxh  +  ah^  ; 
h 

passing  to  consecutive,  values,  we  have, 

^  =  dax^  ;      and      ^dx  =  Sax^dx, 
dx  dx 

In  the  first  example  P  =  a,  and  P'  =  0.  In  the 
second,  P  =  Sax-,  and  P'  =  Saxh  +  ah^. 

15.  Equation  (6)  aifords  the  means  of  determining  the 
differential  coefficient,  and  the  differential  of  any  function, 
whose  form  is  developed  in  terms  of  the  independent 
variable. 

If  we  divide  both  members  of  Equation  (6)  by  the  in- 
crement A,  and  then  pass  to  the  limiting  ratio,  we  have  the 
differential  coeflicient.  If  we  then  multiply  the  differential 
coeflicient  by  the  differential  of  the  independent  variable, 
we  have  the  differential  of  the  function. 

Equal  functions  have  equal  differentials 

16.  If  two  functions,  u  and  v,  dependent  on  the  same 
variable  a,  are  equal  to  each  other,  for  all  possible  values 
of  ar,  their  differentials  will  also  be  equal. 


BEC.    I.]  F  I  R  S  T      P  K  I  N  C  I  P  L  E  S .  25 

For,  X  being  the  independent  Yariable,  we  have  (Art.  14), 
u'  -u  =  Ph-\-  P'h\ 
v'  -  V  =    Qh+  Q'h\ 

in  which  P  is  the  differential  coefficient  of  w,  regarded 
as  a  function  of  a;,  and  Q  the  differential  coefficient  of 
V,   regarded   as  a  function   of  x. 

But,  since  u'  and  v'  are,  by  hypothesis,  equal  to  each 
other,   as  well  as  u  and  v,   we  have, 

Fh  +  PVi2  =   Qh-\-  Q'h\ 

or,   by   dividing  by  h  and  passing  to   consecutive  values, 

P  =  Q, 

.  die        dv 

,  du  T  du  y 

and,  -rr-  dx  =  -7-  ^> 

that  is,  the  differential  of  w  is  equal  to  the  differential 
of  V, 

Converse  not  true. 

17,  The  converse  of  this  proposition  is  not  generally 
true ;   that  is, 

Jf  tico  differentials  are  equal  to  each  other^  we  are  not 
at  liberty  to  conclude  that  the  functions  from  which  they 
were  derivecl,  are  also  equal. 

For,  let  u  =  V  ±  A (1.) 

in  which  ^  is  a  constant,  and  u  and  v  both  functions 
of  X,      Giving  to  x  an  increment  A,   we   shall   nave, 

u'  =  v'  ±:  A^ 


26  DIFFEEENTIAL      CALCULUS.  [SEC.    I. 

from  which   subtract  Equation   (1),   and  we  obtain, 

u'  —  u  =  v'  —  v^ 
and,   by  substituting  for  the   difference    between    the  two 
srtates  of  the   function,   we   have, 

PA  +  r/i?  =    Qh  -f  Q'h\ 

Dividing  by  A,  and  passing  to  consecutive  values,  we 
obtain, 

P=   Q;     that  IS,     ^  =  ^; 

hence,  -j  dx  =  -j-dx;      or,       du  =  dv. 

Hence,  the  differentials  of  u  and  v  are  equal  to  each  other, 
although  V  may  be  greater  or  less  than  w,  by  any  constant 
quantity  A ;  therefore. 

Every  constant  quantity  connected  with  a  variable  by 
the  sign  plus  or  mi7ius,  will  disappear  in  the  differen- 
tiation. 

The  reason  of  this  is  apparent ;  for,  a  constant  does  not 
increase  or  decrease  with  the  variable;  hence,  there  is  no 
ultimate  or  last  difference  between  two  of  its  values;  and 
this  ultimate  or  last  difference  is  the  differential  of  a 
variable  function.  Hence,  the  differential  of  a  constant 
quantity  is   equal  to   0. 

18.    If  we  have  a  function  of  the  form, 

u  =  Av, 

in  which  u  and  v  are  both  functions  of  aj,  and  give  to 
X  an  increment  h,   we  shall  have, 

u'  —  u  =  A[y'  —  ?;), 
or,  Ph  +  rh^  =  A^Qh^  Q'h% 


SEC.    I.]  FIKSTPEINCIPLES.  27 

Dividing  by  A,   and  passing  to   the   consecutive   values, 

P  =  AQ, 

or,  .  Pdx  =  A  Qdx, 

But,  du  =  Pdx,         and        dv  =    Qdx ; 

hence,  du  =  Adv;    that  is, 

T/ie  differeiitial  of  the  product  of  a  constant  hy  a 
variable  qxtantity^  is  equal  to  the  constant  multiplied  hy 
the  differential  of  the  variable. 

Signs  of  the  diflferential  coefficient. 

19.  If  u  is  any  function  of  a;,  and  we  give  to  x  an 
increment   A,   we   have, 

-F  =  ^  +  ^^; 

and  since  h  is  positive,  the  sign  of  the  first  member  "will 
be  positive  when  w  <  w' ;  that  is,  when  ic  is  an  inci'easing 
function  of  x  (Art.  7).  It  will  be  negative  when  w  >  w'; 
that  is,  when  m  is  a  decreasing  function  of  x.  Passing 
to  consecutive  values,  we  have,  under  the   first  supposition, 

-IT-   =   —  P.    under  the  second;   hence, 
V  dx 

The  differential  coefficie?its  have  the  same  sigii  as  the 
functions^  when  the  functions  are  increasing,  and  con- 
trary signs,  when  they  are  decreasing. 

If  we  multiply  by  dx,  we  obtain  the  differentials,  which 
have  the  same  signs  as  the  differential  coefficients. 


28  DIFFEKKNTIAL      CALCULUS.  [SEC.  I. 

Nature  of  a  differential  coefficient,  and  of  a  differential. 

20.  The  method  of  treating  the  Differential  Calculus, 
adopted  in  this  treatise,   is  based  on  three  hypotheses: 

1st.    That  the  independent  variable  changes  uniformly: 

2d.  That  in  changing  from  one  state  of  value  to 
another,  it  passes  through  all  the  intermediate  values;  and, 

3d.  That  any  function  dependent  upon  it,  undergoes 
changes  determined  by  the  equation  expressing  the  rela- 
tions between  them ;  and  that  such  equation  preserves  the 
same  general  form. 

If  the  independent  variable  changes  uniformly,  and  as- 
sumes all  possible  values  between  the  limits  jc  =  a,  and 
X  =  a',  we  have  seen  that  the  change  caimot  be  denoted 
by  a  number.  If,  then,  we  denote  this  change  by  dx,  we 
mean  that  dx  is  smaller  than  any  number ;   hence, 

J_   _ 
dx 

■Ox  1         dx        dx^        dx^      jp 

But,  —  =  —  =  - —  =  >    &G. 

dx       dx^      dx^        dx* 

that  is,  any  power  of  dx  divided  by  a  power  of  dx 
greater  by  1,  is  infinite;  hence,  any  power  of  dx  is  infin- 
itely small,  compared  with  the  power  next  less.  Hence,  it 
follows : 

1st.  That  the  addition  of  dx  to  any  number,  can  make 
no  alteration  in  its  value ;  and  therefore,  when  connected, 
with  a  numeral  quantity  by  the  sign  db  ,  may  be  omitted 
without   error ;   thus, 

Zax  -\-  dx  =  Zax. 


SEC.    I.]  FTUSTPRINCIPLES.  29 

2d.  Since  dx^  ie  infinitely  small,  compared  with  dx ;  that 
is,  wjinitely  less  than  dx,  we  have, 

5ax^dx  -j-  dx^  =  bax^dx\ 

and   similarly  for   the   higher   powers   of  dx. 

The  quantities,  dx^  dx^^  dx\  &c.,  are  called  infinitely 
small  quantities^  or  infinitesimals  of  the  firsts  second,  and 
third  orders  :  from  their  law  of  formation,  it  follows  that. 

Every  infinitely  small  quantity  may  he  omitted  without 
error  when  connected  hy  the  sign  ±  with  any  of  a  lower 
order. 

Hate  of  change. 

21.  The  measure  of  a  quantity,  great  or  small,  is  the 
number  of  times  which  it  contains  some  other  quantity 
of   the   same   kind,    regarded   as   a   unit    of   measure. 

In  the  Differential  Calculus,  dx,  the  differential  of  the 
independent  variable,  is  the  unit  of  measure.     The  rate  of 

dtf 

change,  in   the  function  y,   is  therefore   expressed  by   -^  , 

and  the   actual   change   corresponding  to   dx,   by 

^-Y-dx  =  dy, 
dx  ^ 

22.  The  equation  of  a  straight  line  is, 

y  =  ax  +  b. 

If  we  take  any  point,  as  M,  whose  co-ordinates  are  y 
and  X,  and  a  second  point  N',  whose  co-ordinates  are  y', 
X  +  h,    and    we   have, 

y'-y^ah',     or,    y-^-JL  ^  a    .     .     (1.) 


30 

that   is, 
JSTR 


DIFFERENTIAL      CALCULUS 


[sec.  I. 


MB 


=  tanfrent  NMR  =  a ; 


and,  passing  to  the  consecutive 
values, 

-~-  =  tanf^eut  a   =z  a (2.) 

The    differential    coefficient    -—,    measures    the  rate  of 

(tic 

increase  of  the  ordinate  y,  when  x  receives  the  incre- 
ment dx ;  and  since  this  value  is  independent  of  x,  the 
rate  will  be  the  same  for  every  point  of  the  line ;  that 
is,  the  rate  of  asce?ision  of  the  line  from  the  axis  of 
abscissas,  is  the   same   at   every  point.      And   since, 

dij 

~  dx  =  dy  —  adx. 

dx 

the  change  in  the  value  of  the  ordinate  will  be  uniform^ 
for  uniform   changes  in  the   abscissa. 

23.  Let  us  examine  an 
equation, 

y  =f(x)  .    .    (1.) 

not  of  the  fii-st  degree. 

Let  us  suppose  the  curv'e 
AMN  to  be  such  that  the 
abscissas  and  ordinates  of  its 
different  points  shall  correspond 
to  all  possible  relations  between  y  and  JC,  in  Equation  ( 1  ). 

We  have   seen   (Art.  13)   that, 

dy 


dx 


=  tan   TSF   =  tan  a ;  hence. 


SKC.    I.]  F  1  R  S  T      P  K  I  X  C  T  P  L  E  S  .  31 

the  rate  of  increase  of  tlie  function,  or  the  ascension  of 
the  curve  at  any  point,  is  equal  to  the  tangent  of  the 
angle  which  the  tangent  line  makes  with  the  axis  of  ab- 
sdssas.  "We  also  see,  that  this  value  of  the  tangent  of 
a,  will  vary  with  the  position  of  the  point  M\  hence  it 
ia   a  function   of  x\  therefore, 

In  every  equation^  not  of  the  first  degree^  the  differ- 
ential coefficient  is  a  function  of  the  independeyit  variable, 

1.  "We  have  seen,  that  when  the  points  M  and  K  are 
consecutive,  the  secant  line,  J/iV,  becomes  the  tangent 
line,  TMS  (Art.  13).  The  hue  MR  is  then  denoted  by 
dx^  and  UN  or  RT^  (for  the  points  N  and  T  then  coin- 
cide), by  dy.  If  we  give  to  the  new  abscissa,  x  -f  dx^ 
an  additional  increment  ffe,  and  suppose  the  correspond- 
ing ordinate,  y  •\-  dy^  to  receive  the  same  increment  as 
before,  viz.  :  dy,  the  extremity  of  the  last  ordinate  will 
not  fall  on  the  curve,  but  on  the  tangent  line,  since  the 
triangles  thus  formed  are  similar;  hence, 

TjT  a  function  be  supposed  to  increase  uniformly  from 
any  assumed  value,  the  differential  coefficient  will  be 
constant,  and  equal  to  any  increment  of  the  function 
divided   by   the  corresponding    increment  of  the  variable. 


Nature  of  the  Differential  Calculus. 

24.    In  every  operation  of  the  Differential  Calculus,  one 
of  two  things  u  always  proposed,  and  someti-nes  both : 


32  DIFFERENTIAL      CALCULUS.  [SEC.    1 

1st.  To  find  the  rate  of  change  in  any  vanable  ftmc- 
tion   when   it   begins    to   change   fiom  any   assigned  value. 

2d.  To  find  the  difference  between  any  two  consecutive 
values  of  the  function.  This  difference  is  the  actual 
change  in  the  function,  produced  by  the  smallest  change 
which   takes  place  in  the   independent  variable. 

The  use  of  the  independent  variable  is  to  furnish  a  unit 
of  measure  for  the  increment  of  the  fimction,  tmd  thus  to 
determine  its  rate  of  change^  as  it  passes  through  all  its 
states  of  value.  This  ratio  can  generally  be  expressed  in 
numbers,  either  exactly  or  approximatively. 

25.  The  increment  of  the  function,  corresponding  to 
the  smallest  increment  of  the  vanable,  being  the  difference 
between  any  two  of  its  consecutive  values,  is  a  quantity  of 
the  same  kiiid  as  the  function,  and  differs  from  it  only  in 
this:  that  it  is  too  small  to  he  exj^ressed  bg  numbers.  The 
differential  of  a  quantity,  therefore,  is  merely  an  element 
of  that  quantity ;  that  is,  it  is  the  change  which  takes  place 
when  the  quantity  begins  to  increase  or  decrease,  from 
any  assumed  value.  When  we  find  this  element,  we  have 
the  differential  of  the  function  ;  and  by  dividing  by  dx, 
we  have  the  differential  coefficient.     Hence, 

All  the  operations  of  the  Differential  Calculus  comprise 
but  two  objects  : 

1.  To  find  the  rate  of  change  in  a  function,  when  it 
passes  from  one  state  of  value  to  another,  consecutive 
with  it. 

2.  To  find  the  actual  change  i?i  tJie  function. 

The  rate  of  change  is  the  differential  coefficient,  and  the 
actual  change,  the  d'fferential. 


SECTION     II. 

DIFFERENTIALS    OF   ALGEBRAIC    FUNCTIONS. 
Differential  of  sum  or  difference  of  Functions. 

26.    Let    r    be    a    function    of   the    algebraic    sum  of 
several  variable  quantities,  of  the   form, 

in   which  y,   2,   and  w,   are    functions   of   the   independent 
variable   x. 
If  we  give  9^  to  x  an  increment  ^,   we  shaU  have, 

u'  —  u  =   (y'  —  2/)  +  (2'  —  2)  —  {w'  —  w)  ; 

hence    (Art.  14), 

u'  -ic  =  (Ph  +  rh^)  +  {Qh-\-  Q'h')  -  {Lh  +  X'A2), 

or,      — ^  =  (P  +  P'h)  +  (C  +  Q'h)  -  (Z  +  iVi), 
and  by  passing  to   consecutive  values, 

multiplying  both  members  by  dx^  we  have, 

~dx  =  Pdx  +  C^ic  -  Zc^o;. 
dx 

But    as    P,    §,   and  X,    are    the    differential  coefficients 


3i  DIFFEEENTIAL      CALCULUS.  [SKC.   n. 

of  y,  2,  and  lo,  each  regarded   as   a   function  of  x;   hence, 

f-'cfo  =  f  <&  +  f  &  -  ^&;   that  i^        . 
ax  dx  ax  ax 

The  differential  of  the  sum  or  difference  of  any  numher 
of  functions^  dependent  on  the  same  variable^  is  equal  to 
tJie  sum  or  difference  of  their  differentials  taken  separately. 

Differential  of  a  product. 

ay.  Let  u  and  v  denote  any  two  functions,  x  the 
independent  variable,  and  h  its  increment;  we  shall  then 
have, 

u'  =  u  +  Ph  +  P'h\     and 

v'  =  v-\-  Qh+  Q'h% 
and,  multiplying, 

it'v'  =  (u-\-  Ph-\-  P'h'')  (v  +  CA  +  Qli^) 

=  uv  ■\-  vPh  +  uQh  +  PQh^  +  &c. ; 
hence, 

-T =  vP  +  uQ  +  terms  containing   A,  h^,   and  hK 

If  now  we  pass  to   consecutive  values,   we  have, 

-dT  =  ^^+«G; 

therefore,     d(uv)  =  vPdx  +  u  Qdx  =  vdu  +  udv ;    hence, 

The  differential  of  the  product  of  two  functions  de- 
pendent on  the  same  variable,  is  equal  to  the  sum  of 
the  products  obtained  by  multiplying  each  by  the  differ* 
ential  of  the  other. 


BEC.  II.]         DIFFERENTIALS      OF      FUNCTIONS.  35 

1.    If  we   divide  by  uv^   we   have, 


that  is. 


d(uv)   __  du       dv  .    . 

uv  u         V 


27ie  differential  of  the  product  of  two  functions,  divided 
hy  the  product,  is  equal  to  the  sum  of  the  quotients 
obtained  hy  dividing  the  differential  of  each  by  its 
function. 

28.     "We   can   easily  determine,   from    the    last    formula, 
the  differential  of  the  product  of  any  number  of  functions. 
For,   put   V  =  ts,    then, 

dv  _  d(ts)   _  dt       ds 

V  -  ~^~5"  -  7"^T  •    •    •    •    ^^'^ 

and  by  substituting   ts  for  v,   in   Equation   ( 1 ),  we  have, 

djufs)  _  du       dt       ds  ^ 
uts      '~    u         t         s  ^ 

and  in   a   similar  manner  we   should  find, 

d(utsr.  .  .  .)        du       dt       ds   ,    dr  - 

-^-7 -'  =   —  +  —  +  —  -h  — &G. 

utsr ....  u         t         s         r 

If,  in   the   equation, 

djuts)  _  du       dt       ds 
uts     ~    w         t         s  * 

we   multiply   by  the  denominator  of  the   first  member,  we 
shall  have, 

d{uts)  =  tsdu  4-  usdt  -f  utds;    hence, 

77ie  differential  of  the  product  of  any  nu7nber  of  funo 
tions,   is  equal    to    the  sum   of  the   products  which  arise 


36  DIFFERENTIAL      CALCULUS.  [^SKC.    U. 

by  multiplying    the    differential  of  each  function    by  tlie 
product  of  all  the   others. 


Differentials  of  Fractions. 

29.    To  obtain  the   differential    of   any  fraction  of   the 

form,    -  • 
V 

u 
Put,  -   =  ty        then,        u  =  tv. 

Differentiating  both  members,  we  have, 

du  =  vdt  4-  tdv; 

finding   the  value   of  dt,   and  substituting  for   t   its  value 
- ,    we   obtain, 

_         du       vdv 
dt  = ^ , 

or,   by  reducing  to   a  common   denominator, 

vdu  —  udv     , 
dt  =  :, ;   hence, 


77ie  differential  of  a  fraction  is  equal  to  the  denom- 
inator into  the  differential  of  the  numerator^  minus 
the  numerator  into  the  differential  of  the  denominator^ 
divided  by   the  square   of  the   de7iominator. 

1.    If   the    denominator    is    constant,     dv  =  0,    and   we 

have, 

vdu        du 


SEC.  II. J         DIFFERENTIALS      OF      FUNCTIONS.  37 

2.    If  the  numerator  is  constant,    du  =  0,    and  we  have, 

,^              udv 
dt  = r-  ; 

and  under  this  supposition,  ^  is  a  decreasing  function  of 
V  (Art.  y) ;  hence,  its  differential  coefficient  should  be 
negative    (Art.    19). 

Differentials  of  Powers. 

29.*  To  find  the  differential  of  any  power  of  a  function. 
First,  take  any  function  7/",  in  which  oi  is  a  positive  Avhole 
number.  This  function  may  be  cous>idcrcd  as  composed 
of  n   factors,  each   equal   to   u.       Hence    (Art.  27), 

diw^)         diuuuu  .  .  .  .)         du       du       du       du 

~ — -  =  — ^^ = 1 1 H +... 

w"  {uuuu .  .  .  .)  U  U  II  u 


But  as  there  are  n  equal  factors  in  the  numerator  of  the 
first  member,  there  will  be  7i  equal  terms  m  the  second ;. 

,                                        diiC^)         ndu 
hence,  -^ — '-  =  ; 

therefore,  d{u^)  =  nW^^^du. 

1.    If  71  is  fractional,   denote  it   by    — ,  ,  and  make,. 

s 

r 

V  =z  w%        whence,        v'  =  u^ ; 

^and   since  r  and   s  are   entire  numbers,   we   shall  have,. 

sv'-'^dv  =  ru^-^du'y 
from  which  we  find, 

dv  =  r  du  = du  ; 

•         16 


88  DIFFERENTIAL      CALCdI.US.  [SEC.  II. 

or,   by  reducing, 

r 
T 1 

dv  =  -  W      dui 

s 

which  is   obtained   directly  from  the  function, 
di^w)  =  7i?^«-i  du, 

T 

by  changing  the   exponent  n  to    -  • 

s 

2.  If  the  fractional  exponent  is  one-half,  the  function 
becomes  a  radical  of  the  second  degree.  "We  will  give 
a   specific  rule  for  this  class   of  functions. 

Let  V  —  w^        or,        v  =   y^; 

then,  dv  =  -u^      du  =  ~u   ^  du  =  — — ; 

2  2  2yw 

that  is, 

The  differential  of  a  radical  of  the  second  degree^  is 
equal  to  the  differential  of  the  quantity  under  the  sign 
divided  hy  twice  the  radical. 

3.  Finally,   if  n  is  negative,   we  shall  have, 

1 
W-"  =  — , 
w» 

from  which  we  have   (Art.  29), 

<?(«-)  =  ii^  = 

^d,   by  reducing, 

d{u-^)  =   —  7iu-^-'^du\    hence, 


1  \         —  d{u*)  _    —  WW"  -  ^  du 


SEC.  n.]         DIFFERENTIALS      OF      FUNCTIONS.  39 

The  differential  of  any  power  of  a  furvction^  is  equal 
to  the  expo7ie?it  multiplied  hy  the  function  raised  to  a 
power  less  one,  multiplied  by  the  differe7itial  of  the  funo- 
tion. 

Formulas  for  dififerentiating  Algebraic  Functions. 

1.  d(a)  =0 (Art.  17.) 

2.  d{ax)         ;=  adx (Art.  18.) 

Z.    d{x  +  y)  =  dx  ■{•  dy (Art.  26.) 

4.  d{x  —  y)  =  dx  —  dy (Ait.  26.) 

5.  d{xy)  =  xdy  +  ydx (Art.  27.) 

6.  43         =?^^ (An.  .9.) 

7.  d{x'*')  =  mx^-'^dx (Art.  30.) 

dx 

8.  d{^/^)        =  ^ (Art.  30—2.) 

r  f       r 

9.  d{x-i)       = x-^-'^dx      .     .     .     (Art.  30 — 3.) 

EXAMPLES, 

Find  the   differentials  of  the  following  functions: 

1.  u  =  ax  —  y,  du  —  adx  —  dy. 

2.  w  =  d^x^  4-2.  du  =   2c^xdx  +  dz. 
S,    u  =  hx^  ^  y^  +  a.  du  =  2hxdx  —  Sy^dy, 

4.    u  =  ax^  —hx^-\-x.  du  =  {2ax  —  Zhx^  -f  \)d^ 

h,    u  =  cy"^  —  jc^  -f  ay\       du  =  2[(c  +  d)ydy  —  xdx.^ 


40  DIFFEliENTIAL      CALCULUS.  [SEC.    O- 

6.  u  =  xi/z.  du  =  yzdx  +  xzdy  +  xydz. 

7.  w  =  y2  __  ^2  —  Mt^,      du  =  2(2/(7y  —  20az*dz.) 

8.  u  =  Sa^aj".  du  =  Sna^x'^-'^dx, 

9.  w  =  —  2axr^  -  5  +  ib^aP,     du  =  2(^  +  Qb^x^jdx. 

10.  M  =  5x^  —  2ay  —  52.  (?i^  =  25ar*c7a;  —  2ady, 

11.  w  =  a"  —  aj3  +  46.  c?i^  =  (wiC*-!  —  Sx^)dx. 

12.  w  =  aa;(ic2  +  sb).  du  =  3a(a;2  +  J)c7a;. 

13.  i^  =  (ic2  +  a)  {x  —  a).       du  =  (Sa;^  —  2ax  +  ci)dx. 

14.  w  =  a-y^s^      c7i^  =  2xy^zhlx  +  2x^z^ydy  +  Zx^yH^dz, 


15. 

u  =  ax-{x^  +  a). 

ti^^^ 

=  «:i'(5.T^  ■\-2a)djc. 

16. 

X 
U    =    -' 

y 

du 

ydx  —  xdy 

17. 

a 

du 

A^aydy 

"*  -  5  -  2y2 

{b-  2yY 

18. 

1 

du 

—  dx 

19. 

u  =  fc-»   =    — . 

du 

—  ndx 

20. 

Find  the   flifferential 

of  11  in  the 

equation, 

Put,    a^  —  x^  z=  y;    then,    w  =   -^/y  ;     ^^^  (Art.  30 — 2), 

du  =  — ^. 
2Vy 


SEC.  II.]         DIFFERENTIALS      OF      FUNCTIONS.  41 

But,     dy  =   —  2xdx',     then,   substituting    for    y    and    dy^ 
their  values,  we  have, 

—  Ixdx  —  xdx 


du  = 


2\/aF-  x^ 


21.    u  =   -^2  ax  +  x^ 
1 


22,    u 


23.    u  = 


r.2  —  0-2 


die 


du  = 


-v/l  -  a;2 


£c  +  yi  -  x^ 
24.    w  =  (a  +  V^)^' 


(7w  = 


y'2aic  +  JC* 

(1  -  =^')^ 

dx 


du  =  3(a+V^^ 
2v^ 


25.    w 


d?M  = 


a^  -  X' 


~  a*  +  a2ic2  _f  jc* 

(a^+«^a?'+a;')<?(«^-a;^)  -  {a'^-x'')d(a*+d'x^+x*) 


or. 


c?w 


2aj(2a*  +  2a^x'^  —  x^)dx 


26.  w  =  VoM^  X  v^+  y^- 


du  = 


27.    w  = 


28.    u  = 


{b^  +  y^)xdx  +  {o?  +  a?^).y^y 


(1  +  »)« 

1  +  a;^ 
1  -  a;2* 


du  = 


(1  +  «)*+> 

4xdx 
(1  -  a;2)2* 


42  DIFFEEEXTIAL      CALClfLUS.  [SEC.    II. 


30.    u  = 


_  yi  4-  g  +  V 1 


yi+x  —  y/l  —X 


du  =      g  +  i/T-"^^)^. 

Differential  of   a  particular  binomiaL 
80.— 1.    Let    w  =  (a  +  &b")*. 
Put    a  +  bx^  —  y\    then,    w  =  y" ;    and  (Art.  30), 

(7z*  r=  my^-'^dy. 
But,   from   the  first   equation, 

dy  =    w^"-^<&; 
substituting  for  y  and   fZy  their  values,   we  have, 
du  =  m7ib{a  +  Ja;")"*-^^"-^^; 

that  is,   to   find   the    differential    of   a    binominal    'iinction 
of  this   form, 

Midtiply  the  exponent  of  the  parenthesis,  into  the  ex- 
ponent of  the  variable  within  the  parenthesis,  into  the  co- 
efficient  of  the  variable,  into  the  bmomial  raised  to  a 
power  less  1,  into  the  variable  within  the  parenthesis 
raised  to  a  power  less  1,  i?ito  the  differential  of  the  va- 
riable. 

Rate  of   change  of  the  Function. 

31.    AYhat  is  the  rate  of  change  in  the  area  of  a  square, 
when  the   side  is   denoted  by  the  independent  variable? 
We  have  seen   (Art.  2l)   that  the  differential  coefficient, 

— ,   denotes  the  rate  of   change    in    the  function  w,   cor- 
dx 


SEC.  II.]      DIFFERENTIALS      OF      FUNCTIONS.  43 

responding  to  the  change  dx^  in  the  vakie  of  x\  and  that 
in  all  equations,  except  those  of  the  first  degree,  this  rate 
will  be  variable^  and  a  function  of  x  (Art.  23). 

Let  X  denote  the  side  of  a  square,  and  u  its  area ;   then, 

u  =  x^,       and        —=  2x: 
dx 

hence,  the  rate  of  change  in  the  area  of  a  square  is  equal 
to  twice  its  side ;  that  is,  if  the  side  of  a  square  is  denoted 
by  1,  the  rate  of  change  in  the  area  will  be  denoted  by  2; 
if  the  edge  is  denoted  by  5,  the  rate  of  change  will  be  10; 
and  similarly  for  other  numbers. 

2.  What  is  the  rate  of  change  in  the  volume   of  a  cube, 
when  its  edge  is  the  independent  variable? 

Let  X  denote   the    edge   of  a   cube,    and   u    its   volume ; 
then, 

u  =  a;',       and       —  =  dx^ ; 
dx 

hence,  the  rate  of  change  in  the  volume,  is  three  times 
the  square  of  its  edge.  If  the  edge  is  denoted  by  1,  the 
rate  of  change  in  the  volume  will  be  denoted  by  3  ;  if  the 
edge  is  denoted  by  2,  the  rate  of  change  will  be  12;  if  3, 
the  rate  will  be  27 ;  and  similarly,  when  the  edge  is  denoted 
by  other  numbers. 

Find  the  rates  of  change  in  the  following  functions: 

3.  w  =  8x*  —  3a;2  —  ^x  -\-  a.  A.    ^2x^  —  6^  —  5. 
What  will  express  the  rate  for 

x=\^         a:  =  2,         a;  =  3? 

4.  u  =  {x^  +  a)  (3a;2  +  b).  A.    15a:*  +  Sx^b  +  Qax, 
Find  the  rate  for, 

a:  =  1,  ar  =  2. 


44  ■    DI  FFEnEXTlAL      CALCULUS.  [SEC,  H. 

What  is  the  rate  for,       x  =  0,      x  =  4,      x  =   —  I? 
e.    u  =z   (ax  +  cc2)2.  A.   2  {ax  +  cc^)  («  -|_  2aj). 

What  is  the  rate  for,       a  =  0,       jb  =  1,       jc  =  3  ? 

X _^ 

What  is  the  rate  for,        jc  =  0,      cc  =  1  ? 

Hence,  to  find  the  rate  of  change  for  a  given  value  of 
the  variable  :  Find  the  differential  coefficient^  and  substi- 
tute  the  value  of  the  varia^Je  in  the  second  member  of  the 
equation. 

Partial   Diflferentials. 

32,     If  we  have   a  function   of  the   form, 

u  =f{x,y) (1.) 

the  equation  denotes  that  u  is  a  fiinction  of  the  two 
variables,  x  and  y.  If  we  suppose  either  of  these,  as  y, 
to   remain   constant,   and  x  to   vary,   we   shall  have, 

|'=/'(-'^) (^•) 

if  we  suppose  x  to  remain  constant,  and  y  to  vary,  we 
shall  have. 

The    differential    coefficients    which    are    obtained    under 
these  suppositions,  are  called  partial  differential  coefficients. 


SEC.  n.]         DIFFERENTIALS      OF      FUNCTIONS.  45 

The  first  is  the  partial  differential  coefficient  with  respect 
to   a*,    and   the   second   with   respect  to   y. 

33.  If  we  multiply  both  members  of  Equation  ( 2 ) 
by  dx^  and  both  members  of  Equation  ( 3 )  by  c?y,  we 
obtain, 

^^  dx  =  f'{x,  y)dx,        and        ^  ^^  =  f'\^^  y)^y' 

The   expressions, 

du  -  du  ^ 

are  called,  pai'tial  differentials;  the  first  a  partial  differ- 
ential with  respect  to  a,  and  the  second  a  partial  differ- 
ential  with   respect   to  y ;   hence, 

A  PARTIAL  DIFFERENTIAL  COEFFICIENT  IS  the  differential 
coefficient  of  a  fimction  of  two  or  more  variables^  under 
the  siipposition  that  only  one  of  them  has  changed  its 
value;  and, 

A  PARTIAL  DIFFERENTIAL  is  the  differential  of  a  func- 
tion of  two  or  more  variables^  under  the  sup2)ositio?i  that 
only  one  of  them  has  changed  its  value. 

If  we  suppose  both  the  variables  to  undergo  a  change 
at  the  same  time,  the  corresponding  change  which  takes 
place  in  u^  is  called,  the  total  differential.  If  we  extend 
this  definition  to  any  number  of  variables,  and  assuma 
what  may  be  rigorously  proved,  viz. : 

That  the  total  differential  of  a  function  of  any  nmnher 
of  variables  is  equal  to  the  sum  of  the  p>artial  differ- 
entials, 


4:0  DIFFERENTIAL      CALCULUS.  [SEC.  II. 

we     have    a    general     formula    applicable    to     every    fimo 
tioii  of  two  or  more  vaiiables.  ^ 

EXAMPLES. 

1.    Let  u  =  ic^  +  2/3  _  25    then, 

—  dx  =  2xdx,        1st  partial  differential , 

uX 

g^y  =  3yV?y,       2cl         " 

^dz   =   -dz,        3d        « 
dz 

hence,  du  =  2xdx  +  3yWy  —  dz. 


2.    Let  u  —  xy\    then, 

J.?x  =  ydx, 

hence,  du  =  ydx  +  xdy. 


3.    Let  u  =  jc'"?/'* ;    then, 

—-dx  =  mx^-'^y^dx^ 

—  fZy  =  wy"  -  ^  ic'"(7y ;    hence, 
cly 

du  =  maf^-'^ydx  -f  ny'^-^x'^dy  =  x^-'^y-^^mydx  +  7ixdy), 


SEC.  II.]         DIFFEKEKTIALS      OF      FUNCTIONS.  47 


4.    Let 


hence, 


u  =  -I    then, 

y 

du  ^         dx 
—  dx—  — , 
dx  y 

du  ^  xdy 

-     _  ydx  —  xdy 


5.    Let    u 


ay 


'X'  +  y 


=   =  ayix^  +  2/^)     2  ;    then, 


du        _  ayxdx 


hence, 


du  _ 


c?t« 


{x^  +  2/2)^ 


1  3  > 

.2     I      -,2\2 


ayxdx  —  ox^dy 


6.    Let  u  =  xyzt\    then, 

e7i«  =  yztdx  +  ccs^cZy  +  xytdz  +  a-yse?^. 


SECTION     III. 

INTEGRATION      AND      APPLICATIONS. 

34.  An  Integral  is  a  functional  expression,  either  al- 
gebraic or  transcendental,  derived  from  a  differential. 

Differentiation  and  Integration  are  terms  denoting 
operations  the   exact   converse   of  each   other. 

Differentiation  is  the  operation  of  finding  the  differ- 
ential timction  from  the  primitive   function. 

Integration  is  the  operation  of  finding  the  primitive 
function   from  the   differential  function. 

Rules  have  been  found  for  the  differentiation  of  every 
form  which  a  function  can  assume.  Hence,  in  the  Differ- 
ential Calculus,  no  case  can  occur  to  which  a  known  rule 
is  not  applicable.  In  the  Integral  Calculus  it  is  quite 
otherwise. 

In  returning  from  a  known  differential  to  the  integral 
from  which  it  may  have  been  derived,  we  compare  the 
differential  expression  with  other  expressions  which  are 
known  to  he  differentials  of  given  functions^  and  thus 
arrive  at  the  form  of  the  integral,  or  primitive  function. 
The  main  operations,  therefore,  of  the  Integral  Calculus, 
consist  in  transforming  given  differential  expressions  into 
others  which  are  equivalent  to  them,  and  which  are  differ- 
entials of  knoTNTi  functions  ;  and  thus  deducing  formulas 
applicable  to   all    similar  forms. 

The    integration    is    indicated    by    placing    the    sign     / 


SEC.   in.]  INTEGRATION.  49 

before  the  expression  to  be  integrated.  It  is  equivalent 
to   "integral   of";   thus, 

J  2xdx  =  x\ 

is  read:   "Integral  of  2xdx^   is  equal  to   a^." 

Integration  of  Monomials. 

35.    The   differential   of  every  expression  of  the   form, 

w  =  ic"»,        is        du  =  mx'^-^dx     (Art.  30), 

which  has  been  found  by  multiplying  the  exponent  into 
the  variable  raised  to  a  power  less  one,  into  the  differ- 
ential of  the  variable. 

If,  then,  we   have  a  differential  expression,  of  the  form, 

mTS^-^dx^     or,    x'^dx, 

we  can  find  its  integral  by  reversing  the  above  rule ;  that 
is,   to   find  the  integral   of  such   an   expression. 

Add  1  to  the  exponent  of  the  variable,   and  then  divide 
hy  the  new  exponent  into   the  differential  of  the  variable.* 

EXAMPLES. 

Find  the  integrals  *of  the  following  differential  expressions : 

1.  If  (7w  =  2xdx,  I  du  =  ^r-  =  x^. 

*J  2  X  dx 

2.  J£  du  =  Zx^dx,  fdu  =   --—  =  x^. 

^  S  X  dx 

• M , 

*  This  rule   applies  to   every   case    of   a    differential  monomial  of  th« 
form,    Ax'^dx,  except  that  in  which   m   is    —  1    (Art.  90). 


50  DIFFERENTIAL      CALCULUS.  [sEC   III. 


3.     If  du  =  oTdx^  Jdu 


oT+^dx  ar+^ 


(m4-  l)dx      m  +  1 


4.  Jf  du  =  X- ^dx,  I  du  =        ^  _     _       ^  „ 

'  J  _  2dx  2aj2 

5.  If  c?w  =  x^^/xdx^  Jdu  =  jx^dx   =      -x^V^- 

36.  We  have  seen,  that  the  differential  of  the  product 
of  a  constant  by  a  variable,  is  equal  to  the  constant  multi- 
plied by  the  differential  of  the  variable  (Art.  18).  Hence, 
the  integral  of  the  product  of  a  constant  hy  a  differ- 
ential, is  equal  to  the  constant  'multi2ylied  hy  the  integral 
of  the   differential;  that  is, 

/ax^dx  =z  a  I  x'^dx  =  a aj"  ^  ^. 
«/                     m  +  1 

Hence,  if  the  €X2yression  to  he  hitegrated  has  one  or 
more  constant  factors,  they  should,  at  once,  he  placed  as 
factors,  without  the  sign  of  the  hitegral. 

37.  It  has  been  shown  that  the  differential  of  the 
sum  or  difference  of  any  number  of  variables  is  equal  to 
the  sum  or  difference  of  their  differentials  (Ai*t.  26). 
Hence,  if  we  have   a  differential  expression  of  the  form, 

du  =  2ax^dx  —  hydy  —  z^dz\  we  may  write, 
J  du  =  lajx^dx  —  hjydy  —Jz^dz'y  or, 

fdu  =  -ax^  ^^2  _^iL;  that  is, 
•/         .  3  2^  3  ' 

The  integral  of  the  algehraic  sum  of  an^f  number  of  dif 
ferentials  is  equal  to  the  algebraic  sum  of  their  integrals. 


6KC.   III.J  INTEGRATION.  ,  51 


Correction — Indefinite— Particular — and  Definite  Integrals. 

3§.  It  has  been  shown  that  every  constant  quantity 
connected  with  a  variable  by  the  sign  plus  or  minus,  clis« 
appears  in   the   differentiation   (Art.  17);  that  is, 

d{a  +  jc*")  =  fZiC"  =  mx'^-'^dx. 

Hence,  the  same  differential  may  have  several  integral 
functions  differing  from  each  other  by  a  constant  term. 
Therefore,  in  passing  from  a  differential  to  an  integral 
expression,  we  must  annex  to  the  first  integral  obtained, 
a  constant  term,  to  compensate  for  the  constant  term 
which  may  have  been  lost  in  the    differentiation. 

For  example,   it  has  been   shown  in  Art.  (22),   that, 

dy 

^  =  a,        or,        dy  =  adx, 

is  the  differential  equation  of  every  straight  line  which 
makes  "wdth  the  axis  of  abscissas  an  angle  whose  tangent 
is   a.      Integrating  this   expression,   we   have, 

J  dy  z=  aj  dx (1.) 

or,  y  =  ax', 

or,   finally,  y  =  ax  +  O (2.) 

If,  now,  the  required  line  is  to  pass  through  the  ongin 
of  co-ordinates,   we   shall  have,   for 

jc  =  0,      2/  =  0,        and  consequently,       (7=0. 

But  if  it  be  required  that  the   line   shall  intersect   the 


52  DIFFER  EX  TIAL      CALCULUS.  [sEC.  111. 

axis   of  Y  at   a   distance   from    the    origin   equal    to    +  J, 
we   shall  have,   for 

X  =  0,      2/  —  +  J,        and        consequently,       (7  =   +  5; 

and  the  true  integral   will  be, 

y  =:  ax  +  b (3.) 

If,  on  the  contrary,  it  were  required  that  the  right  line 
should  intersect  the  axis  of  ordinates  below  the  origin, 
we  should  have,   for 

X  =  Oj        y  =   —  J,        and  consequently,         C  =  —  b; 

and  the   true   integral  would  be, 

y  =  ax  —  b (4.) 

The  constant  (7,  which  is  added  to  the  first  integral, 
must  have  such  a  value  as  to  render  the  functional  equa- 
tion true  for  every  possible  value  that  may  be  attributed 
to  the  variable.  Hence,  after  having  found  the  first  integral 
equation,  and  added  the  constant  C,  if  ice  then  make  the 
variable  equal  to  zero,  the  value  which  the  function  assumes 
will  be  the  true  value  of  (7. 

1.  An  indefijiite  integral  is  the  first  integral  obtained, 
before   the  value   of  the   constant   C  is   determined. 

2.  A  particular  integral  is  the  integral  after  the  value 
of  C  has  been   found. 

3.  A  definite  integral  is  the  integral  correspondino*  to 
a  given   value   of  the   variable. 

Thus,  Equation  (2)  is  an  indefinite  integral,  because,  so 
long  as    G   is  undetermined,  it   will  be   the   equation   of  a 


SEC.  III.]  INTEGRATION.  53 

system  of  parallel  straight  lines.  Equations  (3)  and  (4) 
are  particular  integrals,  because  each  belongs  to  a  par- 
ticular line. 

Origin  of  the  Integral. 

39.  The  origin  of  an  integral  function  is  its  zero  value. 
The  value  of  the  variable  corresponding  to  the  oriizin  of 
the  integral,  is  found  by  placing  the  second  member  ot 
the  equation  expressing  the  particular  integral,  equal  to 
zero,  and  finding  therefrom  the  value  of  the  vaiiable. 
Thus,  if  in  Equation  (3),  we  make  y  =  0,      we  have, 

ax  +  b  —  0^       and       x  ~  —  -^ 

a 

"which  shows  that  the  origin  of  the  function  y  (that  is 
y  =:  0),  is  on  the  side  of  negative  abscissas,  and  at  a  dis- 
tance from  the  origin  equal  to  —  -*     In   Equation  (4),  it  is 

at  a  point  whose  abscissa  is  -• 

a 

Integration  between  limits. 

40.  Having  found  the  indefinite  integral,  and  the  par- 
ticular integral,  the  next  step  is  to  find  the  definite  in- 
tegral; and  then,  the  definite  integral  between  given  limits 
of  the  variable. 

Let  us  take  the  particular  integral  found  in  Equation  (3),, 

y  —  ax  ■\-  h. 

If  it  is  required  to  find  the  value  of  the  function  y,  for- 
a  given  value  of  the  variable  sc,  as,  x  =  x\  y  will  be- 
come a  constant  for  this  value,  and  w^e  shall  have, 

y'  =  ax'  ■\-h (5.) 

which  is  a  definite  integral.- 


54 


DIFFEKEXTIAL      CALCULUS, 


[sec.  III. 


^^' 


M 

f- 

a 

y 

0 

p    « 

If  we  wish  the  value  of  the  function  corresponding  to  a 
second  abscissa,    x  =  x'\   we  shall  have, 

y"  =  ax"  +  5 (6.) 

If  we  subtract  Equation  ( 5  )  from  Equation  (  6 ),  we  have, 

y"  -y'  =  a{x"-x')  ....     (7.) 

which  is  the  definite  integral  of  y,  taken  between  the  lim- 
its,  X  =  x\   and  x  =  x" . 

If;   x'  =  OF,   and  x"  =  OQ;  then, 
y'    =  PM,  and  y''  =  QJST;  hence, 
y"  -y'  =  a{x"  -  x')  =  NB ; 

Therefore :    The  integral  of  a  func- 
tion, taken  between   tioo   limits,   indi- 
cated by  given  values   of  x,  is  equal  to   the  difference  of 
the  definite  integrals  corresponding  to  those  limits. 

Let  us  now  explain  the   language  employed  to   express 
these  relations.      The  modified  form  of  Equation  ( 1 ), 

J(dy)^^^  =  ajdx, 
is  read:   "Integral  of  y,   when  x  is  equal  to  jb';"    and 

J(dy)^^Tr  =  ajdx, 
is  read:  "Integral  of  y,  when  x  is  equal  to  a";"   and 

05" 

j(dy)  =  afdx, 

IS  read :  Integral  of  the  diiferential  of  y,  taken  between 
the  limits,  x'  and  a;";  the  least  limit,  or  the  limit  correspond- 
ing to  the  subtractive  integral,  being  placed  below. 


SBC.  in.]  INTEGRATION, 


EXAMPLE. 


1.  What  is  the  integral  of  du  =  Qx^dx,  between  the 
limits  a?  =  1,  and  cc  =  3,  if  in  the  primitive  function 
u  reduces  to   81,   when    aj  =  0. 

Jdu  =    fdx^dx  =  dx^  +  C ;    hence, 

fdu  =  3aj3  +  C. 

But  from  the  primitive  function,  w  =  81,  when  aj  =  0; 
hence,     (7  =  81,     and, 

fdu  =3x3+81 (1.) 

/W«,-i  =       3   +  81   =     84    .     .     (2.) 

/(^w)«,-8  =     81    +81   =  162    .     .     (3.) 

s 

f(du)  =  162   -  84  =     78   .     .     (4.) 

What  is  the  value  of  the  variable  corresponding  to  the 
origin   of  the  integral  (Art.  39)  ? 

Making  the   second    member  of  Equation  ( 1 )   equal    0, 

3a;3  +  81  =  0,         or,         x  =  —  3. 

Integration  of  particular  binomials. 
41,   To  integrate  a  differential  of  the  form  (Art.  30), 
du  =  (a  +  bar)V'dx (1.) 


56  DIFTERENTIAL      CALCULUS.  [SEC.  m. 

The  characteristic  of  this  form  is,  that  the  exponent  of 
the  variable  icithout  the  parenthesis  is  less  by  1  than  the 
exponent  of  the  variable  within. 

Put,     {a  +  bx"")  =  z ;        then,     (a  +  bx^'Y  =  g*" ;     and, 

dz 
nbx'^-'^dx  =  dz  j      whence,      a^-Wa;  =  — : ;    hence, 

7ib 


Jdu  =f{a  +  bx'^Yx^'-^dx  =f-^ 


^dz  2'"+! 


7ib  (m  +  l)nb' 

and  consequently, 

_   (c^  +  bx-Y  +  ^ 

Hence,   to   find  the  integral  of  the   above  form, 

1.  If  there  is  a  constant  factor^  j^/ace  it  without  the 
sign  of  the  integral^  and  omit  the  pmcer  of  the  variable 
loithout  the  parenthesis'  and  the  differential: 

2.  Augment  the  exponent  of  the  parenthesis  by  1,  and 
then  divide  this  quantity^  loith  its  exponent  so  increased^ 
by  the  exponent  of  the  parenthesis^  into  the  exponent  of 
the  variable  within  the  parenthesis,  into  the  coefficient  of 
the  variable. 

EXAMPLES. 

1.  f(a  4-  Sxyxdx  =  ^-^~  +  (7;    and 

2.  fm{a  +  bx'^yxdx  =  ~(a  +  ^^)^  +  0. 
mn{a  -  icx'Yx^dx  =  -  —(a  -  4«c*)^  +  C. 


SEC.  ni.]  INTEGRA  r  ION.  57 


Integration  by  Series. 

42.     The   approidmate    integral   of   any  function   of  the 

form, 

du  z=  JCdx, 

may  be  found,  when  ^  is  such  a  function  of  x,  that  it 
can  be  developed  into  a  series.  Having  made  the 
development  of  the  function  -X",  in  the  powers  of  x, 
by  the  Binomial  Formula,  we  multiply  each  term  by  dx,  and 
then  integrate  the  terms  separately.  When  the  series  is 
converging,  we  readily  find  the  approximate  value  of  the 
fimction  for  any  assumed  value  of  the  variable. 

EXAMPLE. 

1.    Find  the  approximate  integral  of, 

in  which,  X  =  (1  -  a;2)"^ 

Developing,   (1  —  x^)    ^,    by  the  binomial  formula,! 

(1  -  a:^^  =  1  +  T'"^  +  Y-  T'^*  +  y4  4*'  +  *"•' 

multiplying  by  dx,   and  integrating,   we  obtain, 

*  Bourdon,   Art.  166.  University,   Art.  32. 
f  Bourdon,  Art.  135.  University,   Art.  104. 


58  DIFFEBBNTIAL      CALCULUS.  [SEC.    m, 

from   which   we    obtain    an    approximate   value    of  w,   cor- 
responding to   any  value  we  may  give   to  x. 


APPLICATIONS     TO     GE0:METEICAL     MAGNITUDES. 

Equations  of  Tangents  and  Normals. 

43.     We    have    seen,    that    if    a?    and    y    denote    the 

co-ordinates    of    every   point   of   a    curve,    —-    will  denote 

the  tangent  of  the  angle   which    the    tangent   line  makes 

with  the  axis   of  abscissas  (Art.  13).      This  value  of   -^ 

^  '  dx 

was  found  under    the    supposition  that  the   second  secant 
pomt  became   consecutive  with  the  first;  hence. 

Any  two  consecutive  points^  must,  at  the  same  time, 
he  in   the  chord,   the  curve,   and  the  tangent. 

Denote  the  co-ordinates  of  the  point  of  tangency,  in  any 
curve,  by  x"  and  y" .  If  through  this  point  we  draw  any 
secant  line,  its  equation  will  be  of  the  fonn, 

1/  -  2/"  =  a{x  -  x'y* 

If  the  second  point  of  secancy  becomes  consecutive  with 
the  first,   we   shall  have   (Art.  13), 

dy" 
hence,  the   equation  of  the  tangent  line   is, 

y  -  2/" -&==-*")  •    •    •    •    ^'-^ 

*  Bk.   I.    Art.  20. 


BBC.  in.]      TANGENTS   AND   NORMALS.  59 

If,  in  the  equation  of  any  curve,  we  find  the  value  of 
-^-f, ,  and  substitute  that  value  in  Equation  (  1 ),  the  equa- 
tion  will  then  denote  the  tangent  to  that  curve. 

1.    By  differentiating  the  equation  of  the  circle, 
{b2  +  2/2  _  7^2^         o'r,        a;"2  _^  y'n  ^  j^^ 


we  have, 


dx"  ~        y"' 


hence,  V  -  y"  =   -  ^,{x-x")', 

or,  by  reducing,         yy"  +  xx"  =  B"^, 

2.  By    differentiating    the    equation    of  the    ellipse,   we 
have, 

dx"  "~       ^y'* 

3.  By   differentiating  the   equation   of  the   parabola,   we 
have, 

dx"  ~  y"'^ 

4.  By   differentiating  the  equation  of  the  hyperbola,  we 

have, 

dy"  _  B'^x" 
dx"  ~  Ahj"' 

Substituting  these  values,  in  succession,  in  Equation  (1), 
and  reducing,  we  shall  find  the  equation  of  the  tangent 
line  to  each   curve. 

•  Bk.  II.   Art.  §.  \  Bk.  III.    Art.    14.  %  ^k.  IV.   Art.  §. 


60  DIFFEKEXTIAL      CALCULUS.  [sEC.  lH. 

44.     The   equation  of  the   normal   is   of  the  form, 

y-y"  =  a'{x-x")    ....     (i.) 

But   since  the   normal  is  perpendicular  to  the   tangent,  at 
the   point   of  contact, 


a 


1  +  aa'  —  0,*       or, 

hence,  the   equation  of  the  normal  is, 

rJx" 


y-y 


dy 


T,(^-^") 


dx" 
dy"' 


.     (2.) 


By  differentiating  the  equation  of  the  circle,  the  ellipse, 
the    parabola,   and    the    hyperbola,    finding  in    each   differ- 

dx" 
cntial    equation    the    value    of ^— ^,     substituting    that 

value    in   Equation  ( 2 ),   and    reducing,  we    shall    find   the 
equation   of  the   normal  line  to   each   curve. 


Value  of  tangent,   sub-tangent,  normal,  and  sub-normal. 

45.  Let  P  be  any  point  of  a 
curve;  TP  the  tangent,  TR  the 
sub-tangent,  PoV  the  normal,  and 
MN"  the   sub-normal. 

Then,  in  the  right-angled  tri- 
angle   TPB, 


PR  =  TRx  tan  PTR  =  TR  x 


<]y. 

dx' 


hence. 


TR  = 


PR 

dy 

dx 


dx 
y-j-  =  Sub-tangent. 


♦  Bk.  I.     Art.  23. 


SEC.  III.]  TANGENTS      AND      NORMALS.  61 

46.     The   tangent    TP  is   equal    to   the   square   root   of 
the   sura   of  the   squares   of  TR  and   PIt\  hence, 


TP  =  yy^l  +^  =  Tangent. 


47.  Since  TPN  is  a  right  angle,  PPJST  is  the  com 
plement  of  TPR\  it  is  therefore  equal  to  PTJR^  and  con- 
sequently its  tangent  is    y- ;   hence, 

PN  =  v-^-  =  Sub-normal. 
^  dx 

48.  The  normal  PN  is  equal  to  the  square  root  of 
the   sum   of  the  squares   of  PP  and  PN'\  hence. 


PiVr  =  y^l  +  g  =  Normal. 

49.     Apply  these  formulas  to   lines  of  the  second  order, 
of  which   the   general   equation   is, 

2/2  —  f)2X  -f  nx'^,* 

Differentiating,   we  have, 

dy         m  -\-  2nx  m  +  2nx 


^^  2y  2y/mx  +  nx^' 

substituting  this  value,   we  find, 

m   =y'^  =  ■'Sl^-^  =  Sul>tangent. 
^  dy  m  +  2rvx  ^ 


-^  V      ^  dy^         V         ^  \m-{-2nx/ 


*  Bk.  V.   Art.  42. 


DIFFERENTIAL      CALCULUS, 


SEC.  UL 


BK  =  y-f   —   — =  Sub-normal. 

^  dx  2 


PN 


=  y  \/l  +  -J^  =  y^^  +  nx'^+  -(m  +  27ixy. 


By  attributing  proper  values  to  m  and  w,  the  aboTe 
formulas  will  become  applicable  to  each  of  the  conic 
sections.  In  the  case  of  the  parabola,  w  =  0,  and  we 
have. 


TR   =  2x, 


BR  = 


m 


TP  =z   -y/mx  +  4a;2, 
PN 


-^' 


mx  -f  -rn^. 


Asymptotes. 

50.  An  asymptote  of  a  curve  is  a  line  which  continually 
approaches  the  curve,  and  becomes  tangent  to  it  at  an 
infinite   distance  from  the   origin   of  co-ordinates. 


Let   AX  and    AY  he.    the 
co-ordinate   axes,  and 


y  -  y 


dy' 


dx 


r,{^-  ^"), 


the    equation    of   any   tangent 
Une,   as    TP. 

If,   in  the  equation  of  the  tangent,  we  make,  in  succes- 
sion, y  =z  Oj    X  =  0,    we   shall  find, . 

dx"  dv" 


&EC,  III.]  ASYMPTOTES.  '  63 

If  the  curve  CPI^  has  an  asymptote  i?^,  it  is  plain 
that  the  tangent  I'T  will  approach  the  asymptote  IiJ5Jj 
when  the  point  of  contact  P,  is  moved  along  the  curve 
from  the  origin  of  co-ordinates,  and  T  and  J)  will  also 
approach  thc;  points  Ji  and  !FJ  and  will  coincide  with 
them  when  the  co-ordinates  of  the  point  of  tangency  are 
infinite. 

In  order,  therefore,  to  determine  if  a  curve  have  asymp- 
totes, we  substitute  in  the  values  of  ^2^  and  AD,  the 
co-ordinates  of  the  point  which  is  at  an  infinite  distance 
from  the  origin  of  co-ordinates.  If  either  of  the  dis- 
tances AT,  AD,  becomes  finite,  the  curre  will  have  an 
asymptote. 

If  both  the  values  are  finite,  the  asymptote  will  be- 
inclined  to  both  the  co-ordinate  axes ;  if  one  of  the  dis- 
tances becomes  finite  and  the  other  infinite,  the  asymptote 
will  be  parallel  to  one  of  the  co-ordinate  axes;  and  if 
they  both  become  0,  the  asymptote  will  pass  through  the 
origin  of  co-ordinates.  In  the  last  case,  we  shall  know 
but   one  pomt  of  the  asymptote,  but  its  direction  may  be 

determined    by  finding    the  value   of   -^ ,    under  the   sup- 
position that  the   co-ordinates  are   infinite. 

51.    Let  us  now  examine  the   equation, 

of  lines   of  the  second   order,  and  see  if  these  lines  have 
asymptotes.      We  find, 

2^/2  _  ^x 


AT  =  X- 


m  4-  2nx        m  +  2nx^ 


64 


AD 


DIFFERENTIAL      CALCULUS 

mx  4-  2?ix'^ 


[sec.  m. 


mx 


2y  2^/m^  4-  7ix'^' 

which   may  be  put  under  the  forms, 


AT  = 


—  4-2/1 

X 


AD  = 


m 


and  making    a;  =   oo,    we  have, 


AR 


AE 


m 


2v^' 


If  now  we  make  w  =  0, 
the  curve  becomes  a  parabola, 
and  both  the  limits,  AE,  AE, 
become  infinite  ;  hence,  the 
parabola  has  no  rectilinear 
asymptote. 

If  we  make  n  negative,  the 
curve   becomes  an  ellipse,  and 

AE  becomes  imaginary ;   hence,  the   ellipse  has  no  asymp- 
tote. 

But  if  we  make   n  positive,   the    equation  becomes  that 
of  the  hyperbola,  and  both  the  values,  AR,  AE,   become 

2B^ 
finite.       If  we   substitute    for    m  its  value,    -7-,    and  fof 

n    its    value    -^,    we   shall  have, 

AR  =   -  A,        and        AE  =  ±  B. 

Hence,   of  the    lines    of  the    second   order,    the    hyperbola 
alone  has  asymptotes. 


«EC.  III.]  EECTIFICATION      OP      CUKVES.  65 

^  Differential  of  an  arc. 

52.  "We  have  seen  that,  when  the  points  which  limit 
any  arc  of  a  curve  become  consecutive,  the  chord,  the 
arc,  and  tangent  become  equal  (Art.  43) ;  therefore,  the 
differential  of  an  arc  is  the  hypothenxise  of  a  right-angled 
triangle  of  which  the  base  is  dx^  and  the  perpendicular 
dy.  Hence,  if  we  denote  any  arc,  referred  to  rectangular 
co-ordinates,  by  z,   we  have, 


dz 


•v/^M-e?j/2.  .  (1.)      or,      z  =  f  ^dx' +  dy'' . .  {2.) 


Rectification  of  a  plane  curve. 

53.  The  rectification  of  a  curve  is  the  operation  of 
finding  its  length;  and  when  its  length  can  be  exactly 
expressed  in  terms  of  a  linear  unit,  the  curve  is  said  to 
be  rectifiahle.      To  rectify  a  curve,  given  by  its  equation: 

Differentiate    the    equation    of  the  curve    and  find    the 

value  of    dy"^    in  terms  of    x    and  dx ;    or  of    dx^    in 
terms  of  y    and   dy^    and  substitute  the  value  so  found 

in    the    differential  Equation    (2).  TJie    second   memher 

will    then    contain    but    one  variable  and  its  differential; 

the  integral  will  express    the    length  of  the  arc  in  terms 
of  that  variable. 

EXAMPLES. 

1.  Find  the  length  of  the  arc  of  a  circle  in  terms  of 
the  radius.  The  equation  of  a  circle  whose  radius  is  1, 
referred    to    rectangular    axes,   when    the   origin   is  at  the 

centre,  is, 

a;2  -f-  y2  _  1^ 


Q^  DIFPEEBNTIAL      CALCULUS.  [SEdH. 

Denoting  the  arc  by  2,  we  have, 

dz  =   ^dx^  4-  dy\         or,        z  =  f-^dx^  +  dy\ 
From  the   equation   of  the   circle,   we  have, 

xdx  +  ydy  —  0 ;        hence,        dy"^  — ; 

1   ■""  X 

r  /77~    x^dx^        r     dx  p^         _i 


dx. 


Developing  the  binomial  factor  into  a  series,  by  the 
binomial  formula,*  multiplying  by  dx,  and  integrating,  we 
have    (Art.  42), 


_1 


la^    .    l.SicS       1.3.5a;7 


(l_x^)  V^  =  .  +  ^  +  _-+__  +  &c.  +  C 

K  we  suppose  the  origin  of  the 
integral  to  be  at  JEJ,  the  correspond- 
ing value  of  X  will  be  zero,  and 
(7=0.  If  now  we  integrate  between 
the  limits  a;  =  0,  and  a;  =  ^,  we 
shall  obtain  the  value  of  the  cor- 
responding arc  in  terms  of  the  radius  1. 

But  X,  or  FM,  is  the    sine    of  the    arc  JEJP^  denoted 
by  z ;   and  when    jb  =  J,    2  =  30° ;  hence, 


30° 


=f(l  -  x^f'^dx  =  1   +  2^3  +  ^.  +  ^<^'^ 


•Bourdon,  Art.  135.    University,  Art.  104. 


SEC.  III.]  RECTIFICATION      OF      CUEVES.  67 

hence, 

„^o     .        ./I    .    l-l-l     .    1-3.1.1         1.3.5.1.1     ,     .    \ 

and  by  taking  the  first  ten  terms  of  the  series,  we  find, 
It  =   3.1415926.  .  .  , 

a  result  true  to  the  last  decimal  figure. 

We  have  thus  found  the  semi-circumference  of  a  circle 
whose  radius  is  1,  or  the  circumference  of  a  circle  whose 
diameter  is  1. 

2.    Find  the  length    of  the    arc    of  a    parabola,   whose 

equation  is, 

2/2  =  2jtXB. 

Differentiating  and  dividing  by  2,   we  have, 

ydy  =  pdx, 


and  consequently, 


«&»=  J<?y»; 


substituting  this  value  in   the   differential  of  the    arc,  we 
have, 

dz  =  ^dy'^^'tdy^ 


developing   the  radical  quantity  by  the  binomial  formula, 
and  integrating  the   terms   separately,   w^e  have, 

_/ll2/3       1111 2/5       113111  y"^  \ 

^-12^-^2*3^  ""2*2*2'5j^  +  2'2*2'2'3*7P~*^'j'^^- 


68 


DIFFERENTIAL      C  A  L  C  U  L  U 


[sec.  in. 


If  we  estimate  the  arc  from  the  principal  vertex,  z  and  y 
will  be  zero  together,  and  C  will  be  zero.  If  we  make 
y  =  p^  z  will  denote  the  length  of  the  arc  from  the  vertex 
to  the  extremity  of  the  ordinate  passing  through  the  focus. 


QUAD  RAT  UKES. 

'54.  QuADKATUEE  is  the  operation  of  finding  the  area 
or  measure  of  a  surface.  When  this  measure  can  be  found 
in  exact  terms  of  the  imit  of  measure,  the  surface  is  said 
to  bo  quadrahle. 

Quadrature    of  plane    figures. 

55,  A  plane  figure  is,  a  portion  of  a  plane,  bounded  by 
lines,  either  straight  or  curved. 

Let  O  be  the  origin  of  a  sys- 
tem of  rectangular  co-ordinates, 
and  oacdeh  any  line  whose  equa- 
tion is  of  the  foiTQ, 


y  =A^) 


(1.) 


r 

c 

d' 



0 

0     A     G     D 


K  the  ordinate  Oo,  denoted  by  y,  move  parallel  to  itself, 
along  OB  as  a  directrix,  and  so  change  its  value  as  always 
to  satisfy  Equation  ( 1 ),  it  -will  generate  the  plane  surface 
oacdehBO^  and  its  upper  extremity  will  generate  the  line 
oacdeh.  The  element^  or  differential  of  this  surface  will  be 
any  one  of  the  trapezoids,  as  CcdD,  when  the  ordinates 
Cc  and  Dd  are  consecutive.  If  we  denote  the  surface  on 
tlic  left  of  the  ordinate  Cc,  by  s,  ds  will  denote  the  area 
of  the  trapezoid.  This  trapezoid  is  composed  of  the  rect- 
angle  Cd\  and  the  triangle  cd'd\  that  is, 

dydx 
ds  =  ydx-^-^' 


SEC.  III.]  QUADRATURES.  69 

But  since  the  product  ydx  is  an  infinitely  small  quantity 
of  the  first  order,  and  dydx  an  iutinituly  small  quantity  of 
the  second  order,  the  latter  may  be  omitted  without  error 
(Art.  20) ;  hence, 

ds  =  ydx;   that  is. 

The  differential  of  a  plane  surface  is  equal  to  the  or- 
dinate into  the  differential  of  the  abscissa. 

To  apply  the  principle  enunciated  in  the  last  equation, 
in  finding  the  measure  of  any  particular  plane  surface: 

Find  the  value  of  y  in  terms  of  r*,  from  tlic  eq'uition 
of  the  hounding  line;  substitute  this  value  in  the  differ- 
ential equation^  and  then  integrate  between  the  required 
limits   of  X. 

Nature    of   the    Integral. 

56.    To  comprehend  the  true  nature  of  an  integral,  we 
must   examine   the   difierential  from  which   it  was   derived*. 
The  differential  of  a  plane  surface  is, 

ds  =  ydx. 

If  we  integrate  between  the  limits  a;  =  0,  and  x  =  OB  =  a,, 
we  write, 


Jds  =  fydx  =  OoacdebB\ 


that  is,  the  first  member  of  the  equation  denotes  the  sum. 
of  all  the  infinitely  small  rectangles  between  the  limiti* 
jc  =  0,   and  X  z=z  a\   the  second  member. 


fydx, 


is  the  same  thing  under  another  form ;   viz. :  it   shows  that 
18 


70  DIFFERENTIAL      CALCULUS.  [SKC.  III. 

eVery  value  of  y,  between  the  limits  y  —  Oo^  and  y  =  Hh^ 
is  imiltiplied,  in  succession,  into  each  base  denoted  by 
dx\  the  sum  of  these  products,  each  of  which  is  ydx^  is 
obviously  the  required  area. 

1.  Perhaps  the  relation  between  the  differential  and  the 
integral,  may  be  more  obvious,  by  observing  the  figure, 
in  which  the  area  is  divided  into  five  parts,  having  equal 
bases.  If  we  bisect  each  base  and  draw  parallel  ordinates, 
we  shall  have  ten  parts;  if  we  bisect  again  and  draw 
parallel  ordinates,  we  shall  have  twenty  parts;  if  again, 
forty;  and  so  on. 

Now,  there  is  no  diflSculty  in  seeing  that  each  bisection 
doubles  the  numbei^  of  parts,  and  diminishes  the  value  of 
each  part ;  and  that  the  sum  of  the  parts  will  be  constantly 
equal  to  the  given  area.  When,  therefore,  each  part  be- 
comes wfinitely  smally  any  fiiiite  number  of  them  is  0; 
but  an  iTifinite  number  is  equal  to  a  finite  quantity,  viz.: 
to  the  given  area. 

Area    of  a   rectangle. 


57.    Let    0  be  the  origin  of  a  J" 

system .  of  rectangular  co-ordinates.  ^ 


A  X 


On  the  axis  of  F,  take  any  dis- 
tance OB  equal  to  h.  Suppose 
the   line    h    to    move    parallel   to 

itself,  along  the  axis  of  -ZJ  as  a  directrix.,  until  it  reaches 
the  position  AC.  During  its  motion,  it  will  generate  the 
rectangle  0(7;  the  foot  of  the  line  will  pass  over  every 
point  in  the  line  OA^  and  the  line  itself  >\ill  occupy  every 
part  of  the  rectangle  OC, 


SEC.  in.]  QUADRATURES.  71 

Since*  the  equation  of  the  line  ^  (7  is, 

y  =  K 

we  shall  have,  for  the  differential  of  the  surface, 

ds  =  hdx. 

Integrating    between    the    limits    jc  =  0,    and   a;  =  5,    and 
observing  that  (7  =  0,  when  a;  =  0,  we  have, 

h 

fds  =  I  hdx  =  hx  =  hb;    that  is, 

0 

27ie  area  of  a  rectangle  is  equal  to  tJie  product  of  its 
base  hy  its  altitude. 

Area    of  a  triangle. 

58,  Let  ABC  be  a  right-angled 
triangle,  and  C  the  origin  of  co- 
ordinates. Denote  the  base  AB  by 
5,  and  the  altitude  GB  by  h.  De- 
note any  line  parallel  to  the  base  by 
y,  and  the  corresponding  altitude 
by  X, 

If  we  suppose  the  base  AB  to  be  moved  towards  the 
vertex  of  the  triangle,  along  CB  as  a  directrix,  and  so  to 
change  its  value,  that. 


h  :  h 


::  2/  :  ic. 


or. 


y 


hx 
h' 


it  is  plain  that  it  wiU  generate  the  surface  of  the  triangle. 
If  we  denote  the  surface  by  5,  we  have, 

ds  =  ydx\ 


72  DIPFEEENTIAL      CALCULUS.  [SEC.  UL 

substituting  for  y  its  value,   and  integrating  between  the 
limits  a;  =  0,  and  a;  =  A,  we  have, 

d,  =  j^Jxdx  =  ^-  =  -; 

0 

that    is,    The    area    of  a    triangle    is    equal   to    half  the 
product  of  the  base  hy  the  altitude. 

Area    of  the   parabola. 

59.  Find   the  area   of  any  portion   of  the   common  para- 
bola whose   equation  is, 

2/2  ::;:  ^yx\        whcuce,        y  =   \/2px. 

This  value  of  y  being  substituted  in  the  differential   equa* 
tion  (Art.  55),  gives  (Art.  36), 

fds  =  f^/2^dx  =  ^fx^dx  =  ?^a;5  4.  C; 

or,     s  =  -^ =  -xy  +  C. 

If  we  estimate  the  area  from  the  principal  vertex,  where 
a;  =  0,  and  y  =  0,  we  have,  (7=0,  and  denoting 
the  particular  integral  by  s\   we   shall  have, 

2 

s'  =  -xy\    that  is, 

o 

The   area    of  any  portion    of  the  parabola^  estimated 

2 
from    the   vertex^   is  equal  to   -    of  the   rectangle  of  the 

abscissa  and  ordinate  of  the   extreme  point.       The  curve 
w,   therefore,  quadkablk. 


SEC.    III.]  QUADRATURES.  73 

1.  To  find  the  area  of  a  parabola  from  the  vertex  to 
the  double  ordinate  through  the  focus.  We  have,  for  these 
limits,   aj  =  0  and  x  =  ip.     Denoting  the  integral  by  s", 

ip 
we  have,  J ds  =  s"  =  ip^, 

0 

which  denotes  the  area  bounded  by  the  curve,  the  axis, 
and  the  ordinate ;  hence,  if  we  double  it,  we  shall  have  the 
required  area;  or, 

2."  =  ip==  1^2  =1(2^)2. 

That  is.  The  area  is  equal  to  one-sixth  of  the  square 
described  on  the  parameter  of  the  axis. 

2.  If  the  area  be  estimated  from  the  ordinate  through 
the  focus,  where  x  =  ip^  and  y  =i?,  G  must  have  such  a 
value  as  to  reduce  the  first  member  to  0:  for,  this  is  the 
origin  of  the  integral. 

We  have,  J ^^  —  f^  +  ^> 

and  for  the  particular  case  of  the  focus, 

fds  =  ^xipxp  +  C=  \p^  +  (7;  hence, 
1^,2+  (7=0;      or,       {7=-Ji?2. 
Hence,  the  integral  from  x  =  \p  Xo  any  value  of  x  is, 

fds  =  -xy  -  -p\ 

Area  of  the  circle. 

60.  The  equation  of  the  circle  referred  to  its  centre 
and  rectangular  axes  is, 


y2  _  7.2  _  a;2 .        Qj.^        y  ^   y?^—  x^ ; 


74  DIFPBBENTIAL      CALCULITS.  [SEC.  IH 

hence,   the   differential  equation  of  the  area   (Art.  57)   is, 

ds  =  (V^-  aJ^^^      ....     (1.) 

in  which  the  origin  of  the  area  is   at  the  secondary  dia- 
meter, where    a;  =  0. 
From  Formula  S^^  page  189,  we  have, 

But,   by  Formula   (13),  Art.  99,  we  have, 

/(r»-.)-^.^=/-^_  =  sin-'?+(7; 


x" 
whence,  by  substitution,  we  have, 

8  =  lx{r^  -  x^y  +  Ir^  sin"*  -  +  C  ,        (2.) 

Estimatmg  the  area  from  the  secondary  diameter,  where 
a;  =  0,    we  have,     (7=0. 

If  we  integrate  between  the  limits  of  jc  =  0,  and 
jB  =  r,  we  shall  have  one  quarter  of  the  area  of  the 
circle.  When  we  make  a;  =  r,  in  Equation  ( 2 ),  the 
first  term  in  the   second  member  becomes   0 ;   and  in  the 

X 

second  term,  -   becomes   1,   and  the  arc  whose  sine  is   1, 
r 

is   90°,  which  is  denoted  by    — ,  to  the  radius   1 ;  hence, 

2 

r 

Jds  =  ^r2  sin-i  1   =  2^'  X  ^;    or, 
Area  of  the  circle   =  ^(o^^  ^  -)  =  r^'K, 


SEC.  m.]  QUADRATURES.  T5 


Area  of  the  ellipse. 

61.     The   equation   of  the   ellipse,  referred  to  its  centre 
and  axes  is, 

^V  ^  ^2a;2  =  ^2jj2  .     iience, 

B 


and  the  differential  equation   of  the  area  is, 
ds  =  ~(A'  -  x^ydx. 

The  second  member  of  this  equation  differs  from  the 
second  member  of  Equation   ( 1 ),  of  the  last  Article,  only 

in   the   constant   coefficient    -r ,    and    the    constant  A"^  for 

A 

r^,    within    the    parenthesis ;    hence,    the    integral    of   that 
expression  becomes  the  integral  of  this,  by  multiplying  it 

by    -J ,    and   changing  r  into  A ;   that  is, 

Area  of  ellipse  =  • — - —    =  A.B.ir;    that  is. 

The  area  of  an  ellipse  is  equal  to  the  product  of  its 
""semi-axes  multiplied  by  -r. 

1.  Let  Q  denote  the  area  of  a  circle  described  on 
the  transverse  axis,  and  Q'  the  area  of  a  circle  described 
on   the   conjugate  axis ;   then, 

^2^  =    §,         and        i?V  =   Q';    hence, 


76  DIFFERENTIAL      CALCULUS.  [SEC.    III. 

^2^2 ^2  ^  QQ^^  and  AB-JT  =  y^Q  x  Q';    that    is, 

The  area  of  an  ellipse  is  a  mean  proportional  between 
the  two  circles  described  on  its  axes. 


0     A     C     D    E    B  X 


QUADRATURE    OF   SURFACES    OF   REVOLUTION. 

62.  Let  oacdeb  be  a  plane 
curve,  0J5  the  axis  of  abscis- 
sas, and  Oo,  Aa,  Cc,  &c.,  con- 
secutive ordinates;  then,  oa,  ac, 
ed,  &c.,  -R-ill  be  elementary  arcs. 
The  surface  described  by  either 
of  these  arcs,  while  the  curve 
revolves  around  the  axis  OB,  will  be  an  element  of  the, 
surface.  We  have  seen,  that  when  the  ordinates  are  con- 
secutive, the  chord,  the  arc,  and  the  tangent,  are  equal 
(Art.  43) ;  hence,  the  surface  described  by  any  arc,  as 
«c,  is  equal  to  that  described  by  the  chord;  that  is, 
equal  to  the  surface  of  the  frustum  of  a  cone,  the  radii 
of  whose  bases  are  Aa  =  y,  Cc  =  y  -\-  dy,  and  of  which 
the  slant  height  ac  =  ^dx^  +  dy^.  Hence,  if  we  denote 
the   surface   by  s,   we   have,* 

ds  =  -^(22/  +  2y  +  2dy)    X   i^dx^ -}-  dy^; 

or,  omitting   2dy    (Art.  20), 


ds  =  ^ry^dx"^  +  dy'^'j    that    is. 

The  differential  of  a  surface  of  revolution  is  eq^ial  to 
the  circumference  of  a  circle  perpendicxdar  to  the  axis.,  int6 
t/ie  differential  of  the  arc  of  the  meridian  curve. 


SEC.  ni.]  SURFAOES      OP      REVOLUTION.  77 

Therefore,  to  find  the  measure  of  any  surface  of  revo- 
lution : 

Find  the  values  of  y  and  dy^  from  the  equation 
of  the  meridian  curve^  in  terms  of  x  and  dx ;  then 
mhstitute  these  values^  in  the  differential  equation^  and 
integrate  between  the  proper  limits  of    x. 


Surface  of  a  cylinder. 

63.  If  the  rectangle  -4(7  be 
revolved  around  the  side  AB^ 
DC  will  generate  the  surface  of 
a   cylinder. 

Since  the  generatrix  is  parallel 
to  the  axis  AB^  its  equation  will 
be, 


y  =  b,        and  hence,         dy  —  0. 

Substituting  these  values  in  the  differential  equation  of 
the   surface,   we  have, 

Jds  =  f2iryx/dx'^  +  dy^  =  f^^bdx  =  lifbx  -h  C. 

If  we  suppose  A  to  be  the  origin  of  co-ordinates, 
'C  =  0,  and  integrating  between  the  limits  x  —  0  and 
X  =  h^    we  have, 

8  =  2bir7i; 

that  is,     The  measure  of  the  surface  of  a  cylitider  is  equal 
to   the  circumference   of  its   base  into   the  altitude. 


78     .  DIFFERENTIAL      CALCULUS.  [SEC.  HI. 

Surface  of  the  cone. 

64.  If  the  right-angled  triangle 
CBA  be  revolved  around  the  axis 
AC^  GB  will  generate  the  convex 
surface  of  a  cone. 

If  we  suppose   C  to  be  the  origin  ^ 

of  co-ordinates,  the  equation  oi  BG 
will  be, 

y  =1  ax,        and        dJy  =  adx. 

Substituting  these  values  in  the   differential  equation  of 
the  surface,   we  have, 

fds  =  Jl'Tiaxy/dx^  4-  aHM  =  J2'iraxdx^  +  a^  +  (7, 

(Art.  35) =  -R-aa^VT-f  a^  +  G, 

Estimating  the   surface  from  the  vertex,  where    a;  =  0, 
we  have,     (7=0,     and 


s  =  tax' 


ViT 


If  we  make    x  =  h  =  AG,   and    BA  =  b,    we  have, 

a  =  -  ,    and  consequently, 
n 


that  is,    The    convex    surface    of  a   cone  is  equal  to   the 
circumference  of  the  base  into  half  the  slant  height. 


«EC.  in.]  8URFACE0F      THE      SPHERE.  79 


Siuface  of  the  sphere. 

65.     To   find    the   surface   of   a    sphere.       The   equation 
of  the  meridian  curve,   referred  to  the   centre,   is, 


a;2  +  y2  =  i22. 

differentiating,   we   have, 

xdx  +  ydy  =  0 ; 

56. 

_                  xdx                ,           _  , 
dy  = ,        and        dy^  = 

Substituting    for    dy^  its  value,   in    the   differential  of  the 
surface,  which  is. 


ds  =  2'jry^dx'^  +  dy^, 
we  have, 

J*ds  =  J'liryd dx^  +  ^<^a;2  r=  J^'r^Edx  =  2ri?a;  +  O. 

if 

If  we  estimate  the  surface  from  the  plane  passing  through 
the  centre,  and  perpendicular  to  the  axis  of  JC,  we  shall 
have, 

s  =  0,       for      X  =  0,        and   consequently,        (7=0. 

To  find  the  entire  surface  of  the  sphere,  we  must  inte- 
grate between  the  limits  x  =  -\-  H,  and  x  =  —  M^ 
^and  then  take  the  sum  of  the  integrals,  without  reference 
to  their  algebraic  signs;  for,  these  signs  only  indicate  the 
position  of  the  parts  of  the  surface  with  respect  to  the 
plane   passing  through  the   centre. 

Integrating  between  the  limits, 

ic  =  0,  and  a;  =   4-  i?, 


so  DIFFEREXTIAL      CALCULUS.  [SEC.  m. 

we   find,  *  =  2'ri22; 

and  integrating  between  the  limits    x  =  0,   and  a;  =  —  i2, 
there    results, 

8    z=    -  2'S'i22 ; 

hence, 

Surface  =  4crjR2  =  2^i2  X  2i2 ; 

that    is,     Equal    to   four    great    circles^    or    equal    to    the 
curved  surface  of  tJie  circumscribing  cylinder. 

1.    The  two   equal  integrals, 

s  =z  2'rri22,         and        «  =   —  2*i22^    ' 

indicate  that  the  surface  is  divided  into  two   equal  parta 
by  the  plane  passing  through  the   centre. 

Surface  of  the  paraboloid. 

66.     To  find  the  surface  of  the  paraboloid  of  revolution. 
Take  the    equation   of  the   meridian   curve, 


y2  =  2ixc, 

which   being   difl^erentiated,   gives, 

dx^y^y,        and        ax^  = 

P" 

Substituting    this  value    of   dx    in    the    differential  of   the 
surface,    (Art.  62),  we  have, 


ds    =    <l^y^l-^yy     =    "^-lydyy^-^.  p^. 


BEC.  III.]         SURFACE      OF      THE      ELLIPSOID.  81 

But  we  have  found   (Art.  41), 


hence, 


/2';r     ,      y 2'n'  3 


s  =  ?^(2/2  +  ^2)l+  a. 


If  we  estimate    the    surface  from    the  vertex,   at  which 
point    y  =  0,    we  shall  have, 

0  =  ?f!  +  (7,        whence,         0=-?^; 

and  integrating  between  the  limits, 

y  =  0,        and        y  =  b, 


we  have, 


« =  ^^[ii>'+p'f-P'i 


Surface  of  the  ellipsoid. 

67.    To  find  the    sui-face  of   an    ellipsoid   described  by- 
revolving  an   ellipse   about  the  transverse   axis. 

The   equation  of  the  meridian   curve  is, 

whence, 

B^  xdx  B       xdx 

^y  ="  - T^-TT  =  - ^ 


A^   y  A  y^—  x^' 


82 


DIPFEEENTIAL      CALCULUl 


[sEa  ni. 


substituting   the    square    of    dy    in  the   differential  of   the 
surface,  and  for  y  its  value, 


|,/3?^r^. 


we  have, 


di  =  iv^^dxy/A*  -  {A^-  S^)3? 


(1.) 


hence,     fds  =  I^^-^VA^  -  B^fdx^J^-^  -  x*. 


Put,     21'-— yGi^  —  B^  —  D^     Q.  constant  quantity; 


and 


A* 


A^  -  B^ 


=  I^,    also  a  constant. 


and  we  have. 


fds  =  Dfdx^^  -  x' 


E   A 


B  J> 


With  C,  the  centre  of  the 
meridian  curve,  and  the  radius 
jK,  describe  a  semi-circle.   Then, 

/  dx  y/^  —  a;2,    is    a    circular 

segment  of  which  the  abscissa 
is  ic,   and  radius   R. 

If,  then,  we  estimate  the  surface  of  the  ellipsoid  from 
the  plane  passing  through  the  centre,  and  estimate  the 
area  of  the  circular  segment  from  the  same  plane,  any 
portion  of  the  surface  of  the  ellipsoid  will  be  equal  to 
the  corresponding  portion  of  the  circle,  multiplied  by  the 
constant  D,      Hence,   if  we  integrate  the  expression, 


SEC.  hl]  cubatuee    op    volxtmbs.  83 

JdXy/m  -  X\ 

between  the  limits  a;  =  0,  and  x  =  A^  we  shall  have 
the  area  of  the  segment  CGFB^  which  denote  by  B\ 
Hence, 

\  surface  ellipsoid  =     Z>  X  Z)';     and 

Surface  -  2D  x  J)'. 

1.    If  we  make    ^  =  ^,   in  Equation  (1),  the  ellipsoid 
becomes   a   sphere,   and  we  have. 


8  =  J^ntEdx  =  liiRx  4-  G. 


If  we  estimate  the  surface  from  the  plane  passing  through 
the  centre,  (7=0,  and  integrate  between  the  limits 
jc  =  0,    and    jb  =  i?,    we  have, 

\  surface  of  sphere  =  2-^7^ ;     hence, 
Surface  =  4-^722. 

CUBATUEE  OF  VOLUMES  OF  REVOLUTION. 

68.  Cubatuee  is  the  operation  of  finding  the  measure 
of  a  volume.  When  this  measure  can  be  found  in  exact 
terms  of  the  measuring  cube,  the  volume  is  said  to  be 
cahdble. 

69.  A  volume  of  revolution  is  a  volume  generated  by 
the  revolution  of  a  plane  figure  about  a  fixed  line,  called 
the   ojxis. 

If  the  plane  figure  OoacdebB,  be  revolved  about  the 
axis  of  X,  it  will  generate   a  volume   of  revolution. 


8i 


DIFFERENTIAL      CALCULUS. 


[sec.  IU. 


0     A     C     D    E    B  X 


Let  us  suppose  the  ordinates  Aa^  Cc^  Dd^  &c.,  to  be 
consecutive.  During  the  revo- 
lution, any  element  of  the  sur- 
face, as,  AacC^  "will  generate 
the  frustum  of  a  cone,  of  which 
the  radii  of  the  bases  are 
Aa  =  2/,  Cc  =  y  -\-  dy^  and 
the  altitude,  AC  z=  dx.  This 
frustum  will  be  an  element  of  the  volume,  and  will  have 
for  its  measure,* 

lb/'  +  (y  +  dyy  +  y{y  +  dy)\dx. 

If  we  denote  the  volume  by  Fi  develop  the  terms 
within  the  parenthesis,  multiply  by  dx^  and  then  reject 
all  the  terms  containing  the  infinitely  small  quantities  of 
the  second  order  (Art.  20),  we  shall  have, 


dV 


xy'^dx. 


The  area  of  a  circle  described  by  any  ordinate  y,  is 
t!y'^\ ;  hence.  The  differential  of  a  volume  of  revolution  is 
equal  to  the  area  of  a  circle  perpendicular  to  the  axis  into 
the  differential  of  the  axis. 

The  differential  of  a  volume  generated  by  the  revolution 
of  a  plane  figure  about  the   axis  of  Y^  is  rrx^dy, 

70.    To  find  the  value  of  V  for  any  given  volume : 

Find  the  value  of  y'^  in  terms  of  a,  fro7n  the  equation 
of  the  meridian  curve;  substitute  this  value  i?i  the  differ- 
ential equatio?i,  and  th^n  integrate  between  the  required 
limits  of  X. 


*  Leg.,  Bk.  VIII.  P.  6. 


f  Leg.,  Bk.  V.  Prop.  16. 


mas.  in. 


CUBATURK      OP      VOLUMES.  85 


KX.VMPLES. 


1.  Find  the  volume  of  a  right  cylinder  with  a  circular 
base,  whose  altitude  is  h  and  the  radius  of  whose  base  is  r. 

We  have  for  the   differential  of  the  volume, 
dV  =  'jry^dx; 
and  since    y  =  r,    we  have, 

fdV  =f-.f~dx', 

integrating  between  the  limits  a;  =  0,  and  cc  =  A, 
h 
fdV  =  V=  'jrr^x  =  irr^h;    that  is, 

0 

The  measure  of  the  volume  of  a  cylmder  is  equal  to 
the  area  of  its  base  multijylied  by  the  altitude,^ 

2.  Find  the  volume  of  a  right  cone  with  a  circular  base,, 
whose  altitude  is  A,  and  the  radius  of  the  base,  r. 

If  we  suppose  the  vertex  of  the  cone  to  be  at  the  origin: 
of  co-ordinates,  and  the  axis  to  coincide  with  the  axis  of" 
abscissas,  we  shall  have, 

y  =  ax,        or,        y  =  |aj,        and        y2  _  ^a;2. 
substituting  tliis  value  of  y"^,  we  have. 


*  liCgcndre,  Bk.  YIII.  Prop.  2 

J.  V 


86  DIFFERENTIAL      CALCULUS.  [SEC.  HI. 

Integrating  between   the   values    ic  =  0,    and    aj  =  A, 
h 
fdV=  r=  cj  J  =  ^r^  X  ^';    that  is, 

0 

The  measure  of  the  volume  of  a  co9ie  is  equal  to  th6 
area   of  the   base  into   one-third  of  the  altitude.* 

3.    To   find   the   volume   of  a  prolate   spheroid,  f 
The   equation   of  the   meridian   curve   is, 


^2 

and  dV  =  '^^2^^^  ~  x^)dx ;    hence, 


r- 'Si-- -%)*"• 


=  3(3^''"'  -  «')  +   <?• 


K  we  estimate  the  volume  from  the  plane  passing 
•through  the  centre,  we  have,  for  ic  =  0,  F"  =  0,  and 
consequently,  (7=0;  and  taking  the  integral  between 
the  limits    a;  =  0,    and    x  =  A^    we  have, 


JdV  =  I'lrB^  X  A ; 


-which  is  half  the  volume ;   consequently,  the  entire  volume, 

2V  =  1^-52  X  2 A 
3 

•  Legendre,  Bk.  VIII.  Prop.  5.        f  Bk.  VL     Art.  37. 


SBC.  in.]  CUBATUKE      OF      VOLUMES.  87 

But,  <^2  expresses  the  area  of  a  circle  described  on  the 
conjugate  axis,  and  2 A  is  the  transverse  axis ;  hence. 

The  volume  of  a  prolate  spheroid  is  equal  to  two-thirdi 
-  of  the  circumscribing  cylinder. 

1.  If  an  ellipse  be  revolved  around  the  conjugate  axis, 
it  will  describe  an  oblate  spheroid,  and  we  shall  have, 

fdV^firx^dy, 

substituting  for  cc^,  and  integrating,  we  have, 

3 
that  is,  two-thirds  of  the  circumscribing  cylinder. 

2.  If  we  compare  the  two  results  together,  we  find, 

oblate  spheroid  :  prolate  spheroid   :  :   A  :  B. 

3.  If  we  make  J3  =  A^  the  ellipsoid  becomes  a  spheje 
whose  diameter  is  the  transverse  axis.      Then, 

2V=  I'^H^  X  D  =  Ir^D^; 
3  o 

that  is.  Equal  to  two-thirds  of  the  circumscribing  cylinder^ 
or  to  one-si^th  of  ^  into  the  cube  of  the  diameter. 

4.  Find  the  volume  of  a  paraboloid.     The  equation  of 
the  meridian  curve  is, 

V  y"^  =  S/xc;    hence, 

dY  =  2ifpxdx^        and        Y=  itp^. 

If  we  estimate  the  volume  from  the  vertex,  C  =  0.  If  wo 
integrate  between  the  limits  a;  =  0,  and  aj  =  A,  and  de- 
signate by  J,  the  ordinate  corresponding  to  the  abscissa 
jc  =  A,  we  have, 


i 

Sft  DIFFEEEXTIAL      CALCULUS.  [SEC.   HI. 

F=  r^pJi^  =  'kIP-   X  -;    that  is, 
equal  to  half  the  cylinder  having  the  same  base  and  altitude. 

Prism    and    Pyramid. 

1.  Let  ABODE  be  any  polygon,  and  FH  a  line  per- 
pendicular to  the  plane  of  the  base.  If 
the  polygon  moA'e  along  the  line  FH^ 
parallel  to  itself,  it  will  generate  a  prism. 
If  we  denote  the  volume  by  F",  the  area  of 
the  base  by  J,  and  the  indefinite  line  HF     ^'^ 

by  »,  we  shall  have,  

dY=  hdx.  A  B 

and,    integrating    between    the    limits   ic  —  0,    and    aj  =  A, 
h 

J  dV  =  fhdx  =  bxx=l}xh. 

0 

2.  If  we  suppose  the  base  so  to  vary,  as  it  moves  along 
the  line  FIT,  as  to  bear  a  constant  ratio  to  the  square  of 
its  distance  from  the  point  F,  it  T\'ill  generate  the  volume 
of  a  pyramid,  of  which  F  is  the  vertex  and  ABODE  the 
base.*  If  we  denote  the  variable  generatrix,  at  any  point, 
by  y,  and  its  distance  from  the  vertex  by  ar,  we  have, 

dY  =  ydx. 
But,  ^  :  y  :  :  A2  ;  2-2 .     hence,     y  =   ^^  x  ic^ ; 

therefore,  dV=   r^  X  x^dx\ 

and  integrating  between  the  limits  a;  =  0,  and  x  —  h^  we  have, 

0 

•  Legendre,  Bk.  VII.  P.  3.  Cor  1. 


SECTION      I  Y. 

SUCCESSIVE      DIFFERENTIALS  —  SIGNS      OF      DIFFERENTIAL     CO- 
EFFICIENTS   FORMULAS     OF     DEVELOPMENT. 

Successive  Differentials. 

71,  If  u  denotes  any  function,  and  x  the  independent 
variable,  we  liave  seen  that  the  differential  coefficient  P, 
is,  in  general,  a  function  of  x  (Art.  23).  It  may  there- 
fore be  differentiated,  and  a  new  differential  coefficient 
will  thus  be  obtained,  which  is  called  the  second  differ- 
ential coefficienU 

In  passing  from  the  function  u  to  the  first  differ- 
ential coefficient,  the  exponent  of  x  is  diminished  by  1,  in 
every  term  where  x  enters  (Art.  30);  hence,  the  relation 
between  the  primitive  function  ic  and  the  variable  a?,  is 
different  from  that  which  exists  between  the  first  difteren- 
tial  coefficient  and  x.  Hence,  the  same  change  in  a?, 
will  occasion  different  degrees  of  change  in  the  primitive 
function  and  in  the  first  dfferential  coefficient. 

The  second  differential  coefficient  will,  in  general,  be  a 
function  of  a,  exhibiting  a  still  different  relation ;  hence, 
a  new  differential  coefficient  may  be  formed  from  it, 
which  may  also  be  a  function  of  x\  and  so  on,  for  suc- 
ceeding differential  coefficients. 
89 


90  DIFFERENTIAL      CALCULUS.  [SEC.  IT. 

If  we   designate  the  successive  differential  coefficients  by 

p,     q,    r,     5,     &c., 
we  shall  have, 

du  dp  da  ^  . 

du  —  pdx^         dp  =  qdx,         dq  =  rdx. 

But  the  differential  of  p  may  be  obtained  by  differ- 
entiating its  value  —  ,  regarding  the  denominator  dx  as 
constant;   we  therefore   have, 

Jdu\         ^  d^u          - 

Atx)  =  ^^'     ^^'     -^  =  ^^5 

substituting  for  dp  its  value,   and  dividing  by  dx, 

d^u  _ 
^  ~  ^' 

The  notation,  dhi,  indicates  that  the  function  u  has  been 
differentiated  twice ;  it  is  read,  second  differential  of  u. 
The  denominator  dx"-^  denotes  the  square  of  the  differ- 
ential of  a?,  and  not  the  differential  of  x^.  It  is  read; 
differential   of  x,   squared. 

If  we   differentiate   the   value   of  g,  we   have, 
•,ld"u\  ^  d^u         _ 

^\-^)  =  ''^'     "■■'     d^  =  ^^' 

honce,  --—  =  r,     &c. ; 

dx^ 


SEC.  IV.J  SUCCESSIVE      DIFFERENTIALS.  91 

and  in  the  same  manner  we  may  find, 

The  third  differential  coefficient,  -v- ,  is  read :  third 
differential  of  w,  divided  by  dx  cubed ;  and  the  differ- 
ential coefficients  which  succeed  it  are  read  in  a  similar 
manner. 

Hence,  the  successive  differential  coefficients  are, 

du  __  dht  _  d?ii  _  dhi  __  . 

^  -  ^'  ^  -  ^'  ^  -  "*'  ^  -  ^'     *^^-' 

from  which  we  see,  that  each  differential  coefficient  is 
derived  from  the  one  that  immediately  precedes  it,  in  the 
game  way  as  the  first  is  derived  from  the  primitive  fimc- 
tion. 

The  differentials  of  the  different  orders  are  obtained  by 
multiplying  the  differential  coefficients  by  the  correspond- 
ing powers  of  dx ;   thus, 

du 

-^  dx   —  1st  differential  of  w, 


d'^u 

-^dx^  =  2d  differential  of  w, 


'Y—  dx^  =  nth  differential  of  w. 


92  DIFFERENTIAL      CALCULUS.  [SEC.  IV 

EXAMPLES. 

1.  Find  the   differential   coefficients  in   the  function, 

u  =  ax^, 

2.  Find  the   differential  coefficients  in  the  function, 

u  =  ax". 
The  first   differential   coefficient   is, 
du 


dx 


nax 


n-\ 


Since    w,   a,    and    dx^    are    constants,  we    have    for  th« 
second   differential   coefficient, 

—  =  n{n-  l)aa;«-2; 

and  for  the  third, 

-^  =  n{n  -  1>  (»  -  2)aic— 3; 


and  for  the  fourth. 


t^ic^ 


=  w(/i  -  1)  {k       2)  Cw  —  3)aaj'»-*, 


SEC.  IV.]  SUCCESSITE      DIFFERENTIALS.  £3 

It  is  plain,   that  when  n  is  a  positive  integral  nuii'btr, 

the  function 

u  =  aa", 

will  have  n  differential  coefficients.  For,  when  n  differ- 
entiations have  been  made,  the  exponent  of  x  m  fJie 
second  member  will  be  0 ;  hence,  the  nth.  differentia?  co- 
efficient will  be  a  constant,  and  the  succeeding  one?  ff-ill 
be  0.      Thus, 


■^  =  n{n-'l){n-2){n-^) a.  1 


d'^  +  hi 


Sign  of  the  first  differential  coefficient- 
»2.     If  we  have  a   curve   whose   equation   is, 

y  =  A^)y 
and   give  to   X  any   increment  h,   we   have    (Art.  13), 

y^-y  _  fix  +  h)  -  fix) 
h        -  h  ' 

and  passing  to  the  consecutive  values, 

<^y 

-~-  =  tan  a. 
ax 

If  we  so  place  the  origin  of  co-ordinates  that  the  curve 
shall  lie  within  the  first  angle,  7i  will  be  positive,  and 
y'  —  y    will    be    positive    at    all    points    where  the    curve 


94 


DIFFIiPwENTIAL      CALCULUS.  [sEC.  IV, 


recedes  fiora  the  axis  of  X,  and  negative  where'  it  ap- 
proaches the  axis;  and  this  is  true  for  consecutive  as  well 
as  for  other  values.  Hence,  the  curve  will  recede  from  the 
axis  of  JC  when  the  first  dijferential  coefficient  is  positive^ 
and  approach  the  axis  when  that  coefficient  is  negative. 

The  general  proposition  for  all  the  angles  and  every 
possible  relation   of  y  and  x,  is  this : 

The  curve  will  recede  from  the  axis  of  X  when  the 
ordinate'  and  first  differential  coefficient  have  the  same 
sign^   and  approach  it  when  they  have  different  signs. 


1.  To  determine  whether  a 
given  curve,  as  ABC^  recedes 
from,  or  approaches  to  the  axis 
of  X,  at  any  point,  as  G : 
Find,  from  the  equation  of  the 
curve,  the  first  differential  co- 
efficient, and  see  whether  it  is 
positive   or  negative. 


2.    If  the    tangent    becomes  parallel    to   the   axis  of  JT 
at   any  point,   as   JB, 

-^  =  tan  a  =  0 ;        hence,        a  =  0. 
dx 


If  the  tangent  becomes  perpendicular  to  the  axis  of  X^ 
at  any  point,   as  A 


dy 
dx 


=  tan  a  =   CO  ;      hence,        a  =  90*= 


SBC.  IV.]  SUCCESSIVE      DIFFERENTIALS, 


95 


Eign  of  the  second  differential  coefficient. 

73.  A  curve  is  convex  towards  the  axis  of  abscissas 
when  it  lies  between  the  chord  and  the  axis;  and  con- 
cavCf  when  the  chord  lies  between  the  curve  and  the  axis. 

1. 


Figures  ( 1 )  and  ( 2 )  denote  two  curves,  the  one  con- 
vex  and  the   other   concave  towards   the   axis   of  JC. 

Let  J?M  be  any  ordinate  of  either  curve,  P'M'  an 
ordinate  consecutive  with  it,  and  P" M"  an  ordinate  con- 
secutive with   F'M\ 

If  we  designate  the  ordinate  PM  by  y,  P'  Q'  will  be 
denoted  by   dy   (Art.  21),   and   we   shall  have, 

FM'  =  y  +  dy, 

and  since  P"M"  is   consecutive  with   P'M\ 

P"M"  =  y  +    c?y  +  %  +  dy) 
=  y  +  Idy  +  d'y. 

MP  +  P"M" 


Since, 
hence, 
and 


MM'  =  M'M"  =  dx,    QM'  = 


QM'=  y-±y-±^^^y^,,j^^^ 


QM'-FM'  =  QP 


d^ 
~2~ 


96 


DIFFERENTIAL      CALCULUS. 


[sec.  it. 


In    the    case    of   convexity^     QM'  >  F'M\     and    then, 
d^y  is    positive. 


In  the  case  of  concavity,  QM'  <  jPJf' ,  and  then, 
d^y  is  negative ;  and  since  dx^  is  always  positive,  the 
second  differential  coefficient  will  have  the  same  sign  as 
the   second   differential  of  y. 

If  w^e  take  the  case  in  which  the  ordinates  are  nega- 
tive, the  second  differential  coefficient  will  stiU  have  the 
same  sign  as  the  ordinate,  when  the  cm've  is  convex, 
and   a   different   sign   when   it  is   concave.       Hence, 

The  second  differential  coefficient  will  have  the  same 
sign  as  the  ordinate  when  the  curve  is  convex  toioards 
the  axis  of  abscissas,  and  a  contrary  sign  when  it  is 
concave. 

1.  The  second  differential  of  y  is  derived  from  dy  in 
the  same  way  that  dy  is  derived  from  y  (Art.  72)  ;  viz.: 
by  producing  the  chord  PP',  and  finding  the  difference  of 
the  consecutive  values  of  F"Q"  and  SQ'\  w^hich  is  JP"S, 

The  co-ordinates  x  and  y  determine  a  single  point  of 
the  curve,  as  P  ;  these,  in  connection  with  dx  and  dy, 
determine  a  second  point,  P',  consecutive  with  the  first ; 
and  these  two  sets  of  values,  in  connection  with  the  sec- 
ond differential  of  y,  determine  a  third  point,  P",  con- 
secutive with  P. 


SEC.  IV.]        SUCCESSIVE      DIFFERENTIALS.  9^ 

Hence,  the  co-ordinates  x  and  y,  and  the  first  and  second 
differential  coefficients,  always  determine  three  consecutive 
points  of  a  curve. 

2.  When  the  curve  is  convex  towards  the  axis  of 
abscissas,  the  tangent  of  the  angle  which  the  tangent  line 
makes  with  the  axis  of  iZj  is  an  increasing  function  of 
X ;  hence,  its  difierential  coefficient,  that  is,  the  second 
differential  of  the  function,  ought  to  be,  as  we  have  found 
H,  positive   (Art.  19). 

When  the  curve  is  concave,  the  first  differential  coeffi- 
cient is  a  decreasing  function  of  the  abscissas;  hence,  the 
second  differential  coefficient  should  be  negative  (Art.  19). 

Applications. 

74L.  The  equation  of  the  circle,  referred  to  its  centre 
and  rectangular  axes,  is, 

a;2  -}-  2/2  _  7^2 .         hence,        ~    = • 

dx  y 

X 

Placing =  0,  we  have,        a;  =  0. 

Substituting  this  value  of  x  in  the  equation  of  the  circle, 
we  have, 

y  =  ±i?; 

hence,  the  tangent  is  parallel  to  the  axis  of  abscissas  at 
the  two  points  where  the  axis  of  ordinates  intersects  the 
circumference. 

K  we  make,       -^  =   —  ?  =   oo ,      we  have,      y  =  0 ; 

substituting  this  value  in  the  equation  of  the  circle, 
SB  =s    :t:  i2 ;     licnce, 


98  DIFFERENTIAL      CALCULUS.  [SEC.  TV. 

the  tangent  is  perpendicular  to  the  axis  of  abscissas  at 
the  points  where  the   axis  intersects  the   circumference. 

1.  For  the  second  differential  coefficient,  we  find, 

^  _    _  ^ 

which  will  be  negative  when  y  is  positive,  and  positive 
when  y  is  negative.  Hence,  the  circumference  of  the 
circle  is   concave  towards  the   axis   of   abscissas. 

2.  If  we  apply  the  same  process  to  the  equation  of 
the  ellipse,  of  the  parabola,  and  of  the  hyperbola,  we 
shall  find  that  the  tangents,  at  the  principal  vertices,  are 
parallel  to  the  axes  of  ordinates ;  that  the  second  differ- 
ential coefficient  and  ordinate,  in  all  the  cases,  except  that 
of  the  opposite  hyperbolas,  have  contrary  signs ;  and  hence, 
aU  these  curves,  except  the  conjugate  hyperbolas,  are  cor^ 
cave  toicards  the  axis  of  abscissas. 

maclaurin's    tiieoeem. 

T5.     Maclauein's  Theorem  explains  the  method  of  de- 
veloping into  a  series  any  function  of  a  single  variable. 
Let  u  denote  any  fimction  of  a,   as,  for  example, 

u  =   (a  +  x)'" (1.) 

It  is  required  to  develop  this,  or  any  other  function  of 
Xy  into  a  series  of  the  form, 

u  =  A  +  Bx  +  Cx^-h  Dx^  +  ^E^  +  &c.     .    .     (2.) 

in  which  A,  B,  C,  i>,  &c.,  are  independent  of  x,  and 
arbitrary  functions  of  the   constants  which   enter  into  the 


SEC.  IV.]  MACLAURIN'S      THEOKEM.  9? 

second  member  of  Equation  ( 1 ).  When  these  coeffi- 
cieAts  are  found,  the  form  of  the  series  will  be  known. 

Since  the  coefficients,  A^  B,  C,  <fcc.,  are,  by  hypothesis, 
independent  of  cc,  each  will  have  the  same  value  for 
85  =  0,  as  for  any  other  value  of  x;  hence,  it  is  only 
necessary  to   determine  them  for    a;  =  0. 

If  we  make  a;  =  0,  in  Equation  (2),  all  the  terms 
in  the  second  member,  after  the  first,  will  become  zero, 
and  the  second  member  will  reduce  to  A,  which  is  what 
the  function  u  becomes  in  Equation  ( 1 ),  when  a;  =  0 
That  value   is  thus  indicated: 

If  we  find  the  successive  differential  coefficients  of  w, 
from  Equation  (2),   we  shall   have, 

^  =  JS  +  2CX+  dJDx'^  +  4^aj3  +  &c 
%r  -  20  +  2.dDx  +  dAMo^  +  &G. 


whence, 


2.3i>  +  2.3.4^  +  &c. 

&c.,               &c. ; 

A  =  («),_o 

~"  Wa;/aj-o 

_  1  p«\ 

-    1.2\(fe'/»,_o 

1    lcPu\ 

~    1.2.3\c&Vb_o 

Ac. 

&c. ; 

100  DIFFERENTIAL      CALCULUS.  [SEC.  IV. 

hence, 

whicli  IS  Maclaurin's  Formula.  In  applying  the  formula, 
we  omit  the  expressions  jc  =  0,  although  the  coefficients 
are  always  found  under  this  hypothesis, 

EXAMPLES. 

1.    Develop    (a  +  aj)*",    by  Maclaurin's  Formula, 
A  =  a*", 
B  =  {  —  \      =  m{a  +  a;)'"-^  =  ma*"-*, 

^    -   2U^^/   ~  1.2         ^""^"^^  -  1.2       "*       ' 

_    _l_/^\   _  m(m-l)(m-2)  , 

^  -   1.2.3Waj3J   -    1         2  3         ^""^^^ 

_  m  (m-  1)  {m  -  2)         , 
^12  3  ' 

&c.,  &c.,  &c. 

Substituting  these  values  in   Equation  (2),   we  have, 

(a  +  xY  =  a"  +  ma'^-^x  +  ^  ^^  "7  ^^a'^-^ar^ 
the  same   result   as   found  by  the   Binomial  Formula. 


SEC.  IV.]  maclaurin's     thkokem.  101 

2.    If  the   function   is   of  the    form, 

u  =  — i-    =  («  +  x)--^  =  a-4l  +  ^)~'. 
a  +  X        ^  \         at 

we  find, 

^  =  -, 
a 

du\  w     .     V    «  1  1 


ji  =  m  ^   ^i^a^x)--=   -7-1— ^ 

\dxJx^O  V     -r    ;  (a  +  ic) 

_    l/^w\  _    —  1   X   —  2((nr  +jC)2f    _    J^ 


a^ 


~    2.3W/«  =  0~  2.3  ~ 

<fec.,  &c.,  &c. 

Substituting  these   values  in   Maclaurin's  Formula, 

1  1        X    ,    x^       a;3       jg5       ic'    ,     - 

+  -o  -  -.  4-  -^  -  -„  4-  &c. 


a  +  X        a       a?       a?       a*       a^       a 

3.  Develop  into  a   series,   the   function, 

w  =  VaM^^  =  «(l  +  ^) 

4.  Develop  into  a  series,   the  function, 


2 
,2\3 


Note.  -ye.  Maclaurin's  Formula  has  been  deraonstratol 
under  the  supposition,  that  in  Equation  ( 2 )  the  coef- 
ficients   are    independent    of  a;,   and    that    the   equation   is 


20 


102  DIFFERENTIAL      CALCULUS.  [SEC.  IV. 

true  for  every  possible  value  that  can  be  attiibuted  to 
X.  If,  then,  the  function  u  becomes  infinite,  when  jc  =  0, 
the  equation  cannot  be  satisfied;  neither  can  it  be,  if 
any  one  of  the  differential  coefficients  becomes  infinite. 
Hence,  any  form  of  the  function  which  produces  either 
of  these  results,  is  excluded  fi'om  the  formula  of  Mac- 
laurin.       The   functions, 

X 

u  =  log  X,  u  =  cot  ic,  u  =  ax^, 

are  examples  of  such  functions.  In  the  first  case, 
u  =  —  00 ,  when  a;  =  0 ;  *  in  the  second,  w  =  oo , 
when  JB  =  0 ;  and  in  the  third,  J5,  and  the  succeeding 
differential  coefficients,   become  infinite,   when    a;  =  0. 


77.  Taylok's  Theorem  explains  the  method  of  develop- 
ing into  a  series  any  function  of  the  sum  or  difference 
of  two   independent  variables. 

78.  Since  the  sum  or  difference  of  two  independent 
variables  may  always  be  denoted  by  a  single  letter,  any 
function   of  the   foim, 

u'  =  f{x  ±  y\ 

may  be   put   under   the   form, 

u'  =  f{z)i        hy  making        z  =  x  ±  y. 

If  we  suppose  z  to  be  the  abscissa,  and  y'  the  or- 
dinate  of  a   curve,  and  give   to    z    an    increment  /<,  z   will 

*  Bourdon,  Art.  235.  University,  Art.  186.  Lcgendre,  Trig.,  Art.  2*2 


SEC.  IV.]  TAYLOE'S      THEOREM.  103 

become  2  +  A.  If  we  pass  to  consecutive  values,  dz  =  dxj 
and 

du'        du'        ^  ,  .  ^        . 

-=-    =  -7-    =  tan  a.     (Art.  13.) 
dz  (aM 

If  we  suppose  a?  to  remain  constant,  and  y  to  receii^e 
the  increment  A,  z  will  again  become  s  +  A,  and  when 
we  pass  to  consecutive  values, 

dv!        du' 

-=-   =  -T-   =  tan  a. 
dz         dy 

Hence,  in  any  function  of  the  sum  or  difference  of  two 
independent  variables^  the  partial  differential  coefficients 
are,  equal  (Art.  32). 

79.   As  an  example,  take, 

u'=  {x  +  yy. 

If  we  suppose  x  to  vary,  the  first  partial  difierential 
coeflScient  is, 

—  =  n(a;  +  y)— J. 

If  we   suppose  y  to  vary,  it  is, 

^  =  w(cc  +  y)--i; 

and  the  same  may  be  shown  for  the  differential  coef- 
ficients of  the  higher  orders. 

80.    If  any  function  of  the  form, 
u'  =  f(x  +  y), 
be  developed   into    a    series,    it    is    plain    that    the    series 


104  DIFFERENTIAL      CALCULUS.  [SEC.  IV. 

must  have  terms  containing  the  variables  x  and  y,  and 
that  the  constants,  which  enter  into  the  given  function, 
must  also  enter  into  the  development.  Let  us  then 
assume, 

J\u')=f(x-\-y):=A-\-By'^+Cy'  +  JDy^  +  &;Q.     (1.) 

in  which  the  terms  are  arranged  according  to  the  ascend- 
ing powers  of  y,  and  in  which  A^  _B,  (7,  2>,  &c.,  are 
independent  of  y,  but  functions  of  a,  and  arbitrary  func- 
tions of  all  the  constants  which  enter  the  primitive  func- 
tion. It  is  now  required  to  find  such  values  for  the  ex- 
ponents «,  Z>,  c,  &c.,  and  for  the  coefficients  A^  B^  (7,  Z>,  &c., 
as  shall  render  the  development  true  for  all  possible  values 
that  may  be  attributed  to  x  and  y. 

In  the  first  place,   there  can  be  no  negative  exponents. 
For,  if  any  term  were   of  the  form. 


it  might  be  written. 


By--, 
B 


and    making    y  =  0,     this    term    would    become    infinite, 
and  we    should  have, 

which   is   absm'd,   since   the  function   of   ic,  which  is    inde- 
pendent  of  y,   does  not  necessarily  become   infinite  when 

y  =  0. 

The  first  term  A,  of  the  development,  is  the  value 
which  the  primitive  function  u'  assumes  when  we  mako 
V  =  0. 


SEC.  IV.J  TAYLOR'S      TnEOKEM.  105 

If  we  designate  this  value  by  u,  we  shall  have, 

u  =  f(x). 

If  we   differentiate  Equation  ( 1 ),   under  the  supposition 
that  X  varies,   the  partial  differential  coefficient  is, 

du'        dA       dB    ^       dC   ,       dB 

and  if  we  differentiate,  regarding  y  as  a  variable,  the 
partial  differential  coefficient  is, 

du' 

■^   =  a^y«-i  +  bCy^-'  +  cBy<^-'^  +  &c.  .  .  (3.) 

But  these  differential  coefficients  are  equal  to  each  other 
(Art.  78) ;  hence,  the  second  members  of  Equations  ( 2 ) 
and  ( 3 )  are  equal.  Since  the  coefficients  are  inde- 
pendent of  2/,  and  the  equality  exists  whatever  be  the 
value  of  y,  it  follows  that  the  corresponding  terms  in 
each  series  will  contain  like  powers  of  y,  and  that  the 
coefficients  of  y  in  these  terms  will  be   equal  *      Hence, 

a  —  1=0,        b  —  1  =  a,        c  —  1=J,    &c., 

and  consequently, 

<z  =  1,  b  =z  2,  c  =  3,    &c. 

Comparing  the  coefficients,  we  find, 

J.        dA  ^        IdB         ^        IdO 

dx  ^  2  dx^  3  dx 

*  Bourdon,  Art.  195.  University,  Art.  178. 


106  riFFEEENTIAL      CALCULUS.  [SEC.  TV. 

Since  we  have  made, 

/[x  +  y)   =  w',  and         f{x)  =  A  =  u, 

we  shall  have, 

du  ^  d^u  ^  d^u 


I 


dx'  1.2dx^'  1.2.3(^' 

and  consequently, 

,  du       ^    d^u  2/2        ^3u    y^     ,     . 

which  is  the  formula  of  Taylor. 

In  thfe  formula,  u  is  what  u'  becomes,  when    y  =  0; 
^'   ^^^*    ^    becomes  when    y  =  0 ;    ^,  what    -^ 

becomes  when    y  =  0;    and  similarly  for  the   other  coet 
ficients. 

1.    Let  it  be  required  to   develop 

u'  =  f{x  +  2/)  =  (ic  +  y)% 

0 
by  this  formula. 

We  find^ 

M  =  IB",      -^  =  n .  JB"  -  S      -^r^  =  7i .  m  —  Ija'*  - ^  +  ifcc. ; 
'      dx  dx^  ^  ' 

hence, 


m'  =  (iB  +  y)*  =  a*  +  wjC'-V  +       \    ^    ^g"-y 
.    w(n  —  1)  (n  —  2)    ^     ,  -        - 


sECTio:Nr    y 


MAXIMA      AND      MINIMA. 

81.  A  MAXIMUM  value  of  a  variable  function  is  greater 
than  the  consecutive  value  which  precedes,  and  the  con- 
secutive value   which   follows  it. 

A  MINIMUM  value  of  a  variable  function  is  less  than 
the  consecutive  value  which  precedes,  and  the  consecutive 
value  which  follows  it. 

If  we  denote  any  variable  function  by  u^  and  the  inde- 
pendent variable  by  cc,  every  relation  between  u  and  k 
will  be  denoted  by  the  co-ordinates  of  a  curve  whose 
equation  is   (Art.  lO), 

u  =  f{x). 

Let  u'  denote  the  consecutive  or- 
dinate which  precedes  w,  and  u"  the 
consecutive  ordinate  which  follows  it. 
Then,   if  w  is   a  maximum, 


u" 


U  >  u\ 


and 


u  >  ic"  ; 


the  curve  therefore  ascends  just  before  the  ordinate  reaches 
a  maximum  value,  and  descends  immediately  afterwards ; 
hence,  at  the  point  of  maximum,  it  is  concave  towards 
the  axis  of  abscissas    (Art.  73). 

Since  the  curve  ascends  just  before  the  ordinate  reaches 
the  maximum  value,  the  first  differential  coefiicient  will 
be  positive ;  and  since  it  then  descends^  the  first  differ- 
ential   coefficient   will    be    negative    immediately   after    the 


108 


DIFFERENTIAL      CALCULUS. 


[sec. 


maximum  value  (Art.  72).  Hence,  at  the  point  of 
maximum  value  of  the  ordmate,  the  first  differential  co- 
efiicient  will  change  its  sign,  and  therefore  passes  through  0. 
Since  the  curve  is  concave  towards  the  axis  of  abscissas, 
the  second  differential  coefficient  is  negative  (Art.  V3) ; 
hence,   the   conditions   of  a   maximum  value   of  u  are, 


du 
dx 


=  0,  and  y^,    negative. 


82.  Denoting  the  consecutive  or- 
dinates,  as  before,  by  u\  u,  u'\  if 
u   is   a   minimum. 


u  <  u\ 


and 


w  <  w'' ; 


the  cui-ve,  therefore,  descends  just  before  the  ordinate 
reaches  a  minimum,  and  ascends  immediately  afterwards; 
hence,  at  the  point  of  minimum,  it  is  convex  towards 
the   axis   of  abscissas. 

Since  the  curve  descends  just  before  the  ordinate  reaches 
the  minimum  value,  the  first  differential  coefficient  wUl  be 
negative;  and  since  it  then  ascends^  the  first  differential 
coefficient  will  be  positive  immediately  after  the  minimum 
value  (Art.  72).  Hence,  at  the  point  of  minimum,  value 
of  the  ordinate,  the  first  differential  coefficient  will  change 
its   sign,   and   therefore   passes  through   0. 

Since  the  curve  is  convex  towards  the  axis  of  abscissas, 
the  second  differential  coefficient  is  positive  (Art.  73) ; 
hence,   the   conditions   of  a  minimum  value   of  «,   are, 


du 


d?u 
dx^ 


and  ^^ ,     positive. 


SEC.  v.] 


maszima    and    minima. 


109 


83,  Hence,  to  find  the  maximum  or  minimum  value 
of  a  function   of  a  single  variable: 

1.  Find  the  first  differential  coefficient  of  the  function^ 
place  it  equal  to  0,  and  determine  the  roots  of  the  equation, 

2.  Find  the  seco7id  differential  coefficieiit^  and  substi- 
tute each  real  root,  in  succession,  for  the  variable  in  tJie 
second  member  of  the  equation;  each  root  which  gives  a 
negative  result,  will  correspond  to  a  maximum  value  of  the 
function,  and  each  which  gives  a  positive  result  will  coT' 
respond  to  a  minimum  value. 


Point  of  inflection. 

§4.  A  POINT  OF  INFLECTION  is  a  poiut  at  which  a  curve 
changes  its  curvature  with  respect  to  the  axis  of  ab- 
scissas. , 

"When  a  curve  is  concave  towards 
the  axis  of  abscissas,  its  second  difi*er- 
ential  coefficient  is  negative  (Art.  "72) ; 
when  it  is  convex,  the  second  differ- 
ential coefficient  is  positive  (Art.  72) : 
therefore,  at  the  point  where  the 
curve  changes  its  curvature,  the 
second  differential  coefficient  changes 
its  sign,  and  consequently  passes 
through  zero. 

In  the  first  figure,  the  second  differential  coefficient, 
at  the  point  M,  changes  from  negative  to  positive ;  in 
the  second,  from  positive  to  negative.  At  the  point  M, 
in  both  figures,  the  first  differential  coefficient  is  equal 
to   0,   and    the    tangent    line    separates    the    two   branches 


110  DIFFEREXTIAL      CALCULUS.  [SEC.  V. 

of  the  ciiiTe.  When  the  second  differential  coefficient  is 
0,  the  ordinate  at  the  point  has  neither  a  maximum  nor 
a  ndnimmn. 

There  are  three  consecutive  points  of  the  curve  which 
coincide  with  the  tangent,  at  the  point  of  inflection.  This 
is  shown  by  the  equality  of  the  co-ordinates  of  the  point 
M  (in  the  curve  and  tangent),  and  of  the  first  and  sec- 
ond  differentials. 

EXAMPLES. 

1.  To  find  the  value  of  x  which  will  render  the  func- 
tion y  a  maximum  or  minimum  in  the  equation  of  the 
circle, 

y  +x    -  M.  ^-        ^, 

making,  —  -   =  0,        gives,        x  =  0. 

The  second  differential  coefficient  is, 

f^  =  -  ^L±^'.         When,    cc  =  0,    y  =  R\ 
dx^  y^  5  >    ./  > 

hence,  ^  =   "  i' 

which  being  negative,  y  is  a  maximum  for  M  positive. 

2.  Find  the  values  of  x  which  render  the  function  y 
a  maximum  or  minimum  in  the  equation, 

y  =  a  ^  hx  -^  x"^.        Differentiating, 


o» 


t=-6  +  2«,       and       ^  =  2; 


dx 


SEC.  V.J  MAXIMA      AND      MINIMA.  Ill 

and  making,  —  5  +  2a;  =  0, 


gives,  X  = 


2 


Since  the  second  differential  coefficient  is  positive,  this 
value  of  X  will  render  y  a  minimum.  The  minimum 
value  of  y  is  found  by  substituting  the  value  of  jc,  in 
the  primitive   equation.      It  is, 

y  =  a--- 

3.    Find  the  value  of  x  which  will  render  the  function 
u  a  maximum  or  minimum  in  the   equation, 

w  =  a*  +  5^ic  —  c^x^. 
J  =  J3_2c^^,        hence,        «  =  ^, 

and  ^3  =   -2«^ 

hence,  the  function  is  a  maximum,  and  the  maximum 
value  is, 

4^2 


^  =  «*  +  zr2 


4.    Let  us  take  the  function, 

u  =  ^a^x^  —  ¥x  +  c^. 
We  find,        ^  =  9a2a;2  _  54         and        jc  =   ±  — 

The  second  differential   coefficient  is. 


112  DIFFERENTIAL      CALCULUS.  [SEC.  V. 

Substituting  the  jjIus  root   of  £C,   we  have, 

which    gives    a    minimum,    and    substituting    the    negative 
root,   we  have, 

£  =  -.«.., 

which   gives  a  maximum. 

The  minimum  value  of  the  function  is, 
u  =  c^  —  — - ; 


and  the  maximum  value. 


9a 


5.    Find  the  values   of  cc,   which    make    u    a    maximum 
or  minimum  in   the   equation. 


^  jc  =  1,   a  maximum. 


u  =  ic^  —  5x*  4-  525^  —  1. 

X  = 

a;  =   3,    a   minimum. 

6.    Find  the  values  of  x,  which    make    u    a   maximum 
or  minimum  in  the   equation, 

u  =z  x^  —  9ic2  -f  15a;  —  3. 

(  aj  =     +  1,    a  maximum. 
Ans,    ] 

[x  =     +  5,    a  minimum. 


SBC.  v.]  MAXIMA      AND      MINIMA.  113 

1.  Find  the  values  of  x,  which  make  u  a  maximmn 
or  minunum  in  the   equation, 

u  z=  x^  —  Sx^  -{-  Sx  -{-  7. 
Ans.    There  is  no  such  value  of  a;,  since  the  second  differ- 
ential coefficient  reduces  to  0,  for   a;  =  1  ;    hence,  only- 
one  condition  of  a  maximum  or  minimum  is  fulfilled.* 

85*  Notes.  1.  In  applying  the  preceding  rules  to 
practical  examples,  we  first  find  an  expression  for  the 
function  which  is  to  be  made  a  maximum  or  minimum. 

2.  If  in  such  expression,  a  constant  quantity  is  found 
as  a  factor,  it  may  be  omitted  in  the  operation;  for  the 
product  will  be  a  maximum  or  a  minimum  when  the 
variable  factor  is   a  maximum   or  minimum. 

3.  Any  value  of  the  independent  variable  which  renders 
a  function  a .  maximum  or  a  minimum,  will  render  any 
power  or  root  of  that  function,  a  maximum  or  minimum ; 
hence,  we  may  square  both  members  of  an  equation  to 
free   it   of  radicals,   before   differentiating. 

8.  To  find  the  maximum  rectangle  which  can  be  in- 
scribed in   a  given  triangle. 

Let  b  denote  the  base  of  the  triangle,  h  the  altitude, 
y  the  base   of  the  rectangle,   and  x  its  altitude.      Then, 

u  =  XT/  =  the  area  of  the  rectangle. 

But,  b  :  h  :  :  y  :  k  —  x; 

bh  ^  bx 
hence,  y  =  ^ , 

*  We  have  limited  the  discussion  to  a  single  class  of  maxima  and 
minima,  viz. :  that  in  •which  the  first  differential  coefficient  of  the  func- 
tion  is   0,   and  the  second  negative  or  positive. 


lU 


DIFFERENTIAL      CALCULUS 


[sec.  V. 


and  consequently, 


hJvx  —  hx^        b,\  ,v 

u  =  . =  j-(hx  -  X'); 


and  omitting  the  constant  factor    — ,    we  may  write, 

u'  =  hx  —  x^ ; 

for,  the  value  of  x,  which  makes  u'  a  maximum,  will  make 
u  a  maximum    (Art.  85)  ;  hence. 


dx 


=  A  —  2a;, 


or, 


A 
2' 


therefore,   the   altitude  of   the,  rectangle    is    equal  to  half 
the  altitude   of  the  triangle;   and  since, 

the  area  is  a  maximum   (Art.  81 ). 

9.  What  is  the  altitude  of  a  cylinder  inscribed  in  a 
given  cone,  when  the  volume  of  the  cylinder  is  a  maxi- 
mum? 

Suppose    the    cylinder    to    be    in- 
scribed,  as  in  the  figure,   and  let 

ABz^a,  BC=h,  AD  =  x,  ED  =  y\ 

then,     BD  =  a  —  x  =  altitude   of 
the  cylinder,   and 

ify^{a  —  x)*  =  volume  =  v  .  .  (1.) 

From    the   similar   triangles  A£JI> 
and   ACB,  we  have, 

*  Legendre,  Bk.  VIII.     Prop.  2. 


BEC.  v.] 


MAXIMA      AND      MINIMA, 


115 


X  :  y  :  :  a  :  b;        whence,        y  = 


bx' 


Substituting  this  value   in  Equation  ( 1 ),   we  have, 

Omitting  the   constant   factor    — -,   we  may  write, 

v'  —  x^{a  —  X) ; 

for,  the   conditions  which   will    make    v'   a  maximum,   will 
also  make   v   a  maximum  (Art.  85). 

By  differentiating,  we  have, 

c  = 


dx 

Placing, 

lax  -  3a;2  =  0, 

we  have, 

2 

a;  =  0,        and        x  =  -a. 

3 

But, 


dx'' 


=  2a  —  6x  =   —  2a. 


Hence,    the   cylinder  is   a   maximum,   when  its   altitude   is 
one-third  the  altitude  of  the  cone. 

10.  What  is  the  altitude  of  a 
cone  inscribed  in  a  given  sphere, 
when  the  volume   is  a  maximum? 

Denote  the  radius  of  the  given 
sphere  by  r,  and  the  centre  by  C, 
Let  A  be  the  vertex  of  the  re- 
quired  cone,   J3D  the   radius   of  its 

base,   which   denote    by  y,   and    denote    the    altitude    AB 
by  X,      Then, 


116 


DIFFEEENTl  A.L      CALCULUS. 


[sec. 


2/2  —  2ra; 


,2  .  * 


and  if  we   denote  the  volume  of  the   cone  by  v, 

V  =  ^'rrx{2rx  -  jc2)   =  i<r(2riB2  -  x^).f 
Omitting  the   constant  factor  ^-r,   we  have, 

-Y-  =  irx  —  Zx^i        hence, 
ax 


4rx  —  3x^ 


0, 


and 


X  =  -r 
3 


that    is,    the    altitude    of  the    cone    is    four-thirds    of  the 
radius. 

11.    What  is  the  altitude  of  a  cone  inscribed  in  a  sphere 


when   the   convex   surface  is  a  maximum  ? 


Ans,    -  r. 
3 


12.  "What  is  the  length  of  the  axis  of  a  maximum 
parabola  which  can  be  cut  from  a  given  right  cone  with 
a  circular  base? 

Let  JBAC  be  a  section  of  the 
cone  by  a  plane  passed  through  the 
axis ;  and  FD  G  a  parabola  made 
by  a  plane  parallel  to  the  element 
J^A. 

Denote  BC  by  5,  AB  by  a,  and 
CU  by  X ;  then,  J3U  =  b  —  x, 
and  J^,  the  common  ordinate  of 
the  circle  and  parabola,  is  equal  to 
-y/bx  —  iC'.J 


*  An.  G.,   Bk.  II.   Art.  4—8.  f  Leg.,  Bk.  VIII.   Prop.  5. 

X  Legendre,  Bk.  lY.    Prop.  23.   Cor.  2. 


SEC.  v.]  MAXIMA      AND      MINIMA.  117 

By  simiJar  triangles,   we   have, 


_  ax         -.^ 

0 


Hence,   the  area  of  the  parabola   (Art.  59)   is. 


2  ax 


u  =  -  -7-  -x/hx  —  JB^. 
3  0 

Omitting  the  constant  factors,  and  remembering  that  the 
same  value  of  x,  which  renders  u  a  maximum,  will  render 
its  square  a  maximum  (Ait.  79),  and  designating  by  w' 
the  new  function,   we  have, 

w'  =  x'^(bx  —  ic^)   =  bx^  —  a?*,         and 

//?/'  '^  s 

-f-  =  3&c2  _  4a;s .  or,  x  =z  ~b;  and  DB  =  AB. 
ax  4  4 

that  is,  the  axis  of  the  maximum  parabola  is  three-fourtlLS 
the   slant   height   of  the   cone. 

13.  What    is    the    altitude    of  the    maximum    rectangle- 
which   can   be  inscribed   in   a  given   parabola  ? 

A71S.    Two-thirds   of  the   axis,. 

14.  ^What   are  the   sides  of  the  maximum  rectangle  in- 
scribed in  a  given  circle  ? 

Ans.    A  square  whose   side  is    ry^. 

15.  A  cylindrical  vessel,  open  at  top,  is  to  contain  a 
given  quantity  :f  water.  What  is  the  relation  between 
the  radius  of  tne  base  and  the  altitude,  when  the  inte- 
rior surface   is  a  minimum? 

A71S.    Altitude  =  i-adins  of  base. 
21 


lis  DIFFERENTIAL      CALCULUS.  [sEC.  V, 

16.  Required  the  maximum  right-angled  triangle  which 
■*an   be   constructed   on    a   given   line,   as  a  hypothenuse? 

Ans.    When  it  is  isosceles. 

17.  Required  the  least  triangle  which  can  be  formed 
by  the  two  radii,  produced,  and  a  tangent  line  to  the 
quadrant   of  a  given   circle  ?        A?is.    When  it  is  isosceles. 

18.  What  is  the  altitude  of  the  maximum  cylinder 
which   can  be   inscribed  in   a   given  paraboloid? 

A97S.    Half  the   axis. 

19.  What  is  the  altitude  of  a  cylinder  inscribed  in  a 
given   sphere  when   its  convex  surface  is   a  maximum? 

Ans.    r-v/2. 

20.  What   is   the   altitude  of   a   cylinder  inscribed  in   a 

given    sphere,   when   its   volume   is   a   maximum? 

2r 

A71S.      ~~P  ' 

21.  Required  the  base  of  the  maximima  rectangle  which 
can  be  inscribed  in  a  given  ellipse  whose  semi-axes  are  A 
and  J3.  A7is.   Ay/2, 

22.  A  rectangular  sheep-fold,  to  C07itai7i  a  given  arca^ 
is  to  be  built  against  a  wall.  Required  the  ratio  of  the 
least  side  to  the  larger,  so  that  the  cost  shall  be  a  min- 
imum. Ans,   2. 

23.  To  circumscribe  a  given  circle  whose  radius  is  r, 
by  an  isosceles  triangle  whose  area  shall  be  a  minimum. 

Ans.    Perpendicular  to  base  =   3r. 


SECTION     yi. 

DIFFERENTIALS   OF  TRANSCENDENTAL  FUNCTIONS. 
Difierentials    of    Exponential    and    Logarithmic    functions. 

86.     An  Exponential  function  is   one    in  which  the  in- 
dependent variable   enters  as  an   exponent;   as, 

u  =  a' (1.) 

If,   in  a  function  of  this  form,  we  give  to  x  an  incre- 
ment h,  we  have, 

u'  =  a'  +  ^  =  a'a>      ....     (2.) 

Subtracting  Equation   ( 1 )   from   ( 2 ),  member  from  mem- 
ber, we  have, 

w'  —  w  =  a'a^  —  a'  —  a'(a^  —  1) ; 

whence,  =  a*  —  1 (3.) 

Put,  a  =  1  +  5,    and  develop  by  the  binomial  formula; 
we  then  have. 


120  DIFFERENTIAL      CALCULUS.  [SEC.  VL 

Substituting    this    value    of   a*,    in    Equation    ( 3 ),    and 
dividing  by  A,  we  have, 

If  we    now  pass    to    consecutive  values,    by  making    h 
numerically  equal  to   0,   we  have, 

du  I       «>'    .    h^       h*-       h^        . 


and  putting  for  h  its  value,    a  —  1,   we  have. 


Denoting    the    second  member  of  Equation   (4)    by  A:, 
we  have, 

—-T-  =  Ar,        or,        du  =  da'  =  a'kdx    .    .     (5.) 

that  is,  the  differential  of  a  function  of  the  fonn  a*,  is 
equal  to  the  function,  iiito  a  constant  quantity  Jc^  de- 
pendent on  a,  into  the  differential  of  the  exponent. 

Relation  between  a  and  1c, 

87.    The  relation   between    a    and    h   is  very  peculiar, 
and  may  be  determined  by  Maclaurin's  Formula, 

»  =  «-  =  W  +  (S>  +  riS)-  +  ryS)- 

+  &c (6.) 


SEC.  VI.]  EXPONENTIAL      FUNCTIONS.  121 

First,  if  we  make  aj  =  0,  the  function  a*  =  I  =  luy 
The  successive  differential  coefficients  are  found  from 
Equation   ( 5  ) ;    viz. : 

S  =  "'*>   ^*  O  =  *' 

^~j  =  -z-  =  da'k  =  a^'k'^dx;    hence, 

£  =  -^'  -^  &)  =  '- 

=  a'^,       and       (g)  =  ^', 


&c^  &c.,  &c. 

Substituting  these  values  in  Equation  ( 6 ),   "we  have, 

If  we    make    a  =  7 ,    we  shall  have, 

I  111 

^  1   ^  1.2  ^  1.2.3  ^        ' 

designating  the  second  member  of  the  equation  by  c,   and 
jemploying  twelve  terms  of  the  series,   we  find, 

e  =  2.7182818.  . ..; 

1 
hence,  a*  =  e,    therefore,    a  =  e*  .  .  (7.) 

Equation   ( 7 )   expresses  the  relation  between  a  and  k. 


122  DIFFERENTIAL      CALCULUS.  [sEC.  VI. 

A  system  of  logarithms,  called  the  Naperian  system, 
has  been  constructed,  whose  base  is,  e  =  2.7182818.... 
This,  and  the  common  system,  whose  base  is  10,  are  the 
only  systems  in  use.  The  logarithms,  in  the  Naperkn 
system,  are  denoted  by  Z,  and  in  the  common  system 
by  log.  "We  see  from  Equation  (7),  that  k  is  the 
Naperian  logarithm  of  the  number  a.  If  we  take  the 
common  logarithms  of  both  members  of  Equation  (7),  we 
shall  have, 

log  a  =  k  log  e (8.) 

The  common  logarithm  of  e  =  log  2.7182818  .... 
=  .434284482  .  .  .  . ,  is  called  the  modulus  of  the  common 
system,  and  is  denoted  by  M.  Hence,  if  we  have  the 
Naperian  logarithm  of  a  number,  we  can  find  the  com- 
mon logarithm  of  the  same  niimher  hy  multiplying  hy 
t/ie  modulus. 

I^  in  Equation   (8),   we  make    a  =  10,   we  have, 

1  =  k\oge\        or,        ^  =  log  e  =  M\ 

that  is,  the  modulus  of  the  common  system  is  also  equal 
to  1,  divided  hy  the  Naperian  logarithm  of  the  common 
base. 

S8.     From  Equation   (5),   we  have, 

du        da*        y  , 
—  =  —  =  /cdx. 
u  a* 

If  we  make  a  =  10,  the  base  of  the  common  system, 
X  =  log  M,     and 

f7u  1         du        ^^ 

XI  k         u 


SEC.  VI.]  LOGARITHMIC      FUNCTIONS.  123 

that  is,  the  differential  of  a  common  logarithm  of  a 
quantity  is  equal  to  the  differential  of  the  quantity  di- 
vided by  the  quantity  into   the  modulus. 

89.  If  we  make  a  =^  e,  the  base  of  the  Naperiau 
system,  x  becomes  the  Naperian  logarithm  of  w,  and  k 
becomes  1 :  see  Equation   (7) ;    hence,    M  =  I  ;    and 

,  du 

dx  =   — ; 
a'' 

that  is,  the  differential  of  a  Naperian  logarithm  of  a 
quantity  is  equal  to  the  differential  of  the  quantity  di- 
vided hy  the  quantity ;  and  in  this  system^  the  modulus 
is   1.  ^ 

90.  Having  found  that  Jc  is  the  Naperian  logarithm 
of  a,   we  have  from  Equation    ( 5  ),  ' 

du  =  a'ladx; 

that  is,  the  differential  of  a  function  of  the  form  a*, 
is  equal  to  the  function^  into  the  Naperian  logarithm, 
of  the  base  «,   into  the  differential  of  the  exponent. 


EXAMPLES. 

1.  Find  the   differential  of  w  =  a*. 

du  =  a'ladx, 

2.  Find  the  differential  oi  u  =  Ix. 

du  =  —  =  aj-Waj. 

X 

Note.    This   case   would   seem    to   admit    of   integration 
by  the   rule   of  Art.  35;   but    that     nile    applies    to    alge- 


32i  11IFFEKENTIAL      CALCULUS.  [sEC.  VI. 

braic    ftmctions    only,    and    this    form    is    derived    from    a 
transcendental   function. 

S.    Find  the   differential   of  w  =  y*. 
lu  =  xly,    hence, 

—   =  jc—  +  lydx',    hence, 
u         y       "^     ' 

by   clearing  of  fractions,   and  reducing, 

du  =  xy'  -  '^dy  +  y*ly  dx\ 

that    is,     equal    to    the    sum    of    the    partial    differentials 
(Art.  32). 

4.  Find  by  logarithms  the   differential  of  w  =  Qty, 

lu  z=  Ix  -\-  ly-y*    hence, 

—  = 1 — -;     and  by  reducing, 

i4>  tij  y 

du  —  ydx  4-  xdy     (Art.  ay). 

5.  Find  by  logarithms  the   differential  of    w  =  -  . 

lu  —  Ix  —  ly  \\   hence,  by  differentiating, 

du         dx       dy  ^   .  ^     . 

—  = ~  ;    and  by  reducmg, 

^^^yJx-_^     (Art.  29). 

*  Bourdon,    Art.  230.    University,  Art.  1§5. 
f  Bourdon,   Art.  231.    University,   Art.  185. 


SEC.  VI.]  LOGARITHMIC      FUNCTIONS  125 

6.    Find  the  differential  of    u  =  l(-^-]. 


,  ladx 

du  = ' 

a^  —  x^ 


r.    Find  the   differential  of    w  =  l(-^=:^=] 

V-i/oM-  xy 


du 


jc(a2  +  x^) 


8.  Find  the   differential  of    u  =  {a'  +  1)\ 

du  =  2a^{a' +  l)ladx, 

d^ I 

9.  Find  the  differential  of    w  =  • 

a'  4-  1 

,  la'ladx  ■ 

du    =: • 

(a-  +  ir 

10.    Find  the   differential   of    u  =  —  =  (-)'• 

X'         \x/ 

DijQferenfial  forms  which  have  known  integrals. 

91.  If  we  have  a  differential  in  a  fractional  fonn/  in 
which  the  numerator  is  the  differential  of  the  denomina- 
tor,  we  know  that  the  integral  is  the  Naperian  logaritlim 
of  the  denominator  (Art.  §9).  It  frequently  happens, 
however,  that  we  have  to  deal  with  fractional  differen- 
tials which  are  not  of  this  form,  but  which,  by  certaui 
algebraic  artifices,  may  be  reduced  to  it.  We  shall  give 
a   few   examples   of  such  reductions. 


J  26  DIFFERENTIAL      CALCULUS.  [SEC.  VI. 

dx 


Form  1. 


,  /: 


Put        jc^  _j_  ^2  _  -y2 .        then,        xdx  =  vdv. 
Add    vc?a5    to   both  members;    then, 

xdx  +  vdx    =  vdx  +  vdv ;     hence, 
(a;  +  tj)(7ic  =  v{dx  +  ^y) ;     whence, 
dx  +  (Zy         f7a;  dx 


«  +  y  V  ViC^  rt  a2 


;    hence, 


/dx  -\-  dv  _     r       dx 
X  +  V     ~  J  .^^T^jT^ 


Bnt  in  the  first  member,   the    numerator    is    the   differen- 
tial  of  the   denominator ;    hence, 

/-^=  =  l{x  +  v)  =  l(x  +  v^^^^). 


Form  2.  /- 


(fa; 


Put         Va;2  ±  ^ax  =  v ;        then,        x"^  ±  2ax  =  v\ 

Adding    a^    to    both  members,   and  extracting  the   square 
root, 

vdv 


X 


±  a  =   y/v^  +  d?- ;        hence,        dx 


and 


ya2~±  2oa;         Vu2  +  a2 


SEC.  VI.]  LOGARITHMIC      FUNCTIONS.  127 

But   from   the   first   form, 


Substituting  for  v  its  value,  and  for    V^+  ^S    ^^^  value, 
r^_dx      _  _  i{x  ±  a  ■\'  ^/tP-  ±  2ajc). 


„  2(2cfa;  2adx 

Form  3.  -r r;        or,        -^^ 5 

^2  _  jg2 »  a;2  —  a2 


2ac?a;                    2ac?a;  <fa;       ^     dx* 

Since,      -^ -„  =   ,      ,      ■    . =  — — -  + 


ci^  —  x^        (ci  -\-  x)  {a  —  x)        a  -\-  X       a  —  x 

rl   dx  dx   \  _    r   dx      x     r   ^^ 

*f  \a  -\-  X       a  —  x}~*^a-\-x       J  a  —  x 

J  x^  —  a?-  \x  ■\-  a) 

(See  Example  6,  page   126.) 

■ 
Form  4.  — 


Put       v^  +  ^^  =  '^  j       whence,       a}  -\-  x^  =:  v^  \     hence, 
SB^  rrz  -y^  _  ^2^    r^nd    JCcZa;  =  t'^^y,        or,         dx  =  • 

X 
*  University,  Art.  180.     (See  Art.  158.) 


12S  DIFFEllENTl  AL      CALCULUS.  [SEC.  VI. 

la 
Multiply   both   members   by    — - —  ;    we   have, 

— 7^-  =       -o T,  =  H—T—]'i    hence, 

/2adx        _     /-v/a^  +  a;2  —  a\ 
a;v^2"^  2,2    ~      \^^2  ^  ^2  ^  ^ 

In  like  manner  we   should  find. 


Form  5.  /*— : 


Put        -  =  v:        then,        aj-^cfe  =  —  ^y;        and 

X 

''"'^       =.  _ZL^  ;    first  fonn. 


=  -'^sA^) 


=  _  ;/i_±_vL±_^') 


SBC.  VI.]  LOGARITHMIC      FUNCTIONS.  129 


TABLE      OF      FORMS. 


1.  ja'ladx  =  a* (Ex.  1.) 

2.  J—=Jdxx-'^z=  Ix (Ex.  2.) 

3.  f(xy'-'^dy  +  y'ly  X  dx)  =  y    .    (Ex.  3.) 
*•  /"7tI=1      =  KaJ  +  V^^~^').     (Form  1.) 
5.  /     ■ f=  =  l(x  ±  a  -h  s/x"  rb  2aa:).     (2.) 

r.  /Jf^         =.;(^).    (Forms.) 

J  01^  —  o?  \x  +  a/      ^  ' 

8.  —===    =  q-^^— =r= ).    (Form  4.) 

9.  /  — ;p^  =  1 1 pr-  I  •  (Form  4.) 

10.  r^a^  =  -  ;(L±vI±Z^).    (5.) 


130 


DIFFERENTIAL      CALCULUS, 


fSEC.  VI. 


EGA 


CIRCULAR      FUNCTIONS 

92.  Let  O  be  the  centre  of  a  circle, 
A  the  origin  of  arc,  and  j5(7,  DH  any 
two  consecutive  ordinates.  Draw  BE 
parallel  to  0A\  draw  the  radius  OB^ 
and    denote    it    by    1.       Denote    the    arc 

AB  by  z,  and  suppose  z  to  be  the  independent  vari- 
able. Then,  BD  wiU  be  the  differential  of  the  arc  AB\ 
ED^  the  differential  of  the  sine,  and  EB  the  differential 
of  the  cosine,  which  will  be  negative,  since  it  is  a  de- 
creasing  function   of  the   arc   (Art.  19). 

93.  Since  the  triangles  OBC  and  DEB^  have  their 
sides  respectively  perpendicular  to  each  other,  they  "will 
be   similar;*  hence, 

OB  :   OC    w   BD  \  DE\    or, 
1     :  cos  2   :  :     dz    :  cf  sin  s ;    whence, 

(?  sin  2  =  cos  zdz (1.) 


94.  Again,         1  :  sin  g  : :  e?3  :  —  c?cos2;    whence, 

<?  cos  s  =  —  sin  2  (72      .    .    .    .    (2.) 

95.  Since, 

cos  2  =  1  —  ver-sin  g,        c?  cos  2  =   —  c?    ver-sin  2 ; 
hence,  d  ver-sin  2  =  sin  2  <7z     .     .     .     .     (3.) 


*  Legendre,  Bk.  IV.     Prop.  21. 


SEC.  VI.] 


CIKCULAE      FUXCTIONS. 


131 


96.     Again,  tan  z  = 


sin  z 
cos  2 


;    hence, 


,  ^                cos  s  ^  sin  s  —  sin  2  c?  cos  2  , . 
rftanz  = -^- (Art.  29). 


Substituting  for  d  sin  z  and  d  cos  2,  their  values  from 
Equations   ( 1 )   and   (  2  ),   we  have, 

dtanz  =  — - (4.)    » 

cos^s  ^     ^ 

By  similar  processes,  we  can  find  the  differentials  of 
the  co-versed-sine,  cotangent,  secant,  and  cosecant,  in  terms 
of  the   other  functions   and   the   differential   of  z. 


97.  Denote  the  sine  of  the  arc  AJ3  by  y,  its  cosino 
by  X,  its  versed  sine  by  v,  and  its  tangent  by  t.  If 
we  regard  each  of  these  as  the  independent  variable,  and 
the  arc  z  as  the  common  function,  and  find  the  values 
of  z  from  Equations  (1),  (2),  (3),  and  (4),  we  shall 
ha\e. 


dz 


When  radius  =  1, 


dz  =   - 
dz  = 
dz  = 


dx 


dv 


'^2v  —  v^ 
dt 


1    -f   «2 


(5.) 
(6.) 

(8.) 


dz 


When  radius  =  r, 
rdy 


dz  = 
dz  = 
dz  = 


rdx 


-v/7'2  —  x^ 
rdv 


■  ■  (9) 


(10.) 


■v/2ro  —  »« 


r'  +  f 


..(11.) 
.  .   (12.) 


132  DIFFEKENlriAL      CALCULUS.  fsEC.  VI. 

The   differential   of   the   arc,   in  terms    of   either   of  the 
other   functions   is   easily  found. 

©5.    The    following    notation    is  employed  to   designate 
an   arc  by  means   of  any   one   of  its  functions. 

sin  -  ^w,  denotes  the  arc  of  which  u  is  the  sine, 
cos-%,  denotes  the  arc  of  which  u  is  the  cosine, 
tan-^w,  denotes  the  arc  of  which  u  is  the  tangent, 

^  &c.,  &c.,  &c. 


■  u 
If  we  denote  the  sine  of  an  arc  by   -,   instead  of  y,   as 

in  Equation   ( 5 ),  we  shall  have, 

u  ^  du  ^  .      ,  w 

y  =.  -.        dy  =  — ,        and        s  =  sm-^-- 

Substituting  these  values  in  Equation  (5),  we  have, 
dz   =    —==r=:z. (13.) 

Denoting  the   cosine  of  the  arc  by   -,   and  making  like 
substitutions  in  Equation  (6),  we  have, 

az  =  _^^ (14.) 

Denoting  the  ver-sine  of  the  arc  by  -  ,   and  making  like 
substitutions  in  Equation  (7),  we  have, 


dz  =  -J^=.       ....     (15.) 
y2a!<— m' 


SEC.  VI.]  CIKCULAK      FU^CTIONS.  133 

Denoting  tl 
Equation  (8), 


Denoting  the  tangent   of  an   arc  by    - ,   we  have  from 


dz  =  ^^, (16.) 

a'  +  w2 


EXAMPLES. 

1.    Differentiate  the   function, 

z  =  cos~i(w-v/l  —  w")  . 
(-  1  +  1u^)du 


dz  = 


2.    Differentiate  the  function, 

dz   = 


2  =  sin-M2wyl  —  2«2J  .        dz  =  -—— . 


3.  Differentiate  the   function, 

,  X  ,  ydx  —  iC(?y 

s  =  tan  - 1  - ,  cZ^  =  ^-— 7^  . 

2/  y^  +  X- 

4.  Differentiate  the  function, 

z  =  cosic  «^°*. 

Make,        cos  jc  =  w,        and        sin  a;  =  y ; 

then,  2  =  wy,       and,         (Art.  90), 

dz  =  uyludy  4-  yuy~'^du; 

also,  <7m  =    —  sin  a  (/it*,     and     t/y  —  cos  a*  dx ; 

22 


134  DIFFERENTIAL      CALCULUS.  [sEC.  VI. 

hence,  dz  =  uv  \ludy  -\-  -  du\ 

I  cos  X  cos  X \ax, 

cos  x/ 

Di£ferential  forms  which  have  known  Integrals. 

99.  Tbe  first  four  equations  in  Art.  92  furnish  us  four 
forms,  by  taking  the  integrals  of  both  members.  Equations 
(5)j  (6),  (V),  and  (8),  are  of  the  Same  form  as  Equations 
(9),  (10),  (11),  and  (12),  except  that  the  radius  is 
1  in  the  tirst  set,  and  r  in  the  second ;  hence,  the  arc 
g,  in  each  equation  of  the  second  set,  is  r  times  as  great 
as   in   the   corresponding   equation    of  the    first   set.* 

Forms  (13),  (14),  (15),  and  (16),  are  modified  forms 
of  (  5  ),  (  6  ),  (  7  ),  and  (  8  ).  Tl;ey  differ  from  them  only 
in  the  symbol  by  which  the  function  of  the  arc  is  denoted. 

TABLE      OF      FORMS. 

1.  /  cos  zdz        =  sin  2  +  (7. 

2,  J  —  sin  zdz   =z  cos  z  +  C, 

8.  / sin  zdz        =  ver-sin z  -{■  C, 

4.  f-^  =tan2+C. 

^  cos^z 

C.                   f-7^=^     =  sm-iy-h^. 
-^-/T-y^ 

*  Leg.,   Trig.    Art.    30* 


SEC.  VI.] 

6. 


10. 


11. 


12. 


13. 


14. 


15. 


16. 


CIRCULAB      FUNCTIONS. 

—  dx 


135 


f  —  di 
f 


yT—  x* 

dv 


V2v  —  v^ 
dt 


=  cos-^oj  +  (7. 


=  ver-sin-i©  +  {7. 


ItTp         =  ^-'t  +  O. 


/ 


rdy 


/: 


—  rdx 
rdv 


J  r^^t2 


I 
I 
f 


du 


Va2_     y2 


—  du 


du 


'\/2au  —  u^ 

/adu 


=  sin-iy  4-  C. 


=  cos- 'a;  +  C. 


=  verHsin-^v  +  C. 


=  tan-i^  4-  C. 


=  sm-i-  +  C. 
a 


=  cos->-  +  C. 


ver-sin-'-  +  C 
a 


=  tan-i-  +  C. 
a 


136  rUFFERENTIAL      CALCULUS.  fsEC.  TI. 


Applications. 

100.  We  may  readily  find  the  relation  between  the 
diameter  and  the  cii*cumference  of  a  circle  from  either 
of  the  first  four  equations  of  Art.  97. 

1.    To  find  this  ratio  from  Equation  (5),  which  is, 

Developing  by  the   Binomial   Formula,   we  have, 
^  =  1  +  22/^  +  2-:^y*  +  2-;^^^  +  &c.;    whence, 

1  1  Q  1  ^  *» 

dz  =dy+  -y^y  +  ^  y*dy  +  —^  y^dy  +  <fec. 

/<fe=2=  y  +  ^y'  +2X52^'  +2S72''  +  *''- 

If  we  make    2  =  80°,    of  which  the   sine,  y  is    - ,  * 
we  have, 

11  1-3        ,        1.3.5        ,     „ 

^^    =  2  +  2:3^3  +  ^^^  +  ^^6.7:2^  +  *^- 

By  multiplying  both  members  of  the  equation  by  6, 
and   taking   twelve  terms   of  the   series,  we   have, 

180°  z=z  <n  -   3.1415924, 
whic-h  is   true   to  the  last   place,    which   should  be    6. 
*  Logendre,  Trig.  Art.   64. 


SEC.  Tl.]  CIRCULAR      FUNCTIONS.  137 

2.    Find  the  ratio   from  Equation   ( 8 ),  which   is, 

<^  =  riV     -     S  =  r^  =  (^  + '^)- 

Developing  by  the  Binomial  Formula,   we  have, 

—  =  1  _  ^2  4-  ^  -  ^6  ^  ^8  _  &c  J    whence, 

dz  —  dt  —  Mt  +  t^dt  —  t^de  +  t^de  —  &c. 

/'  t^       t^       t"^       t^ 

dz  =  z  =  tm-'^t  =  ^  —  3  +  5  —  y  +  9  ""  <^c- 

This  series  is  not  sufficiently  converging.  To  find  the 
value  of  the  arc  in  a  more  convei-ging  series,  we  employ 
the  following  property  of  two   arcs,   viz. : 

^our   times    the    arc    ichose    tangent    is    - ,   exceeds   tJie 

5 


arc  of  45°   hy   the  arc  whose  tangent  is 


239 


*  Let    a    denote    the    arc    whose  tangent  is   — .     Then,   Leg.,  Trig. 

5 


Art.  36., 

2  tan  a  5 

tan  2a  =     • : — r-  =  z-x 

1  —  tan'a  12 


2  tan  2a  120 

tan  4a  =  Y::ri^^*2~a  "  \19' 


The  last  number  being  greater  than  1,  shows  that  the  arc  4a  ex- 
ceeds 46*.      Making, 

Aa  =  A,  45°  =  B, 


138 


DIFFEEE2^TIAL      CALCULUS.  [sEC.  VI. 


But,        tan-'  Q  =  i-  -  ^  +  A^  _  -L  +  Ac, 

-.f_L\  =  J i_  +  _» ^_  + 

\239/         239        3(239)3  ^  5(239)^        7(239)'  ^ 

f     4^1  _  J_   4.  J L  4.\ 

V  5        3.53  "^  6.55       7.7^  "^/ 

4JL -^—  +_^ I— A 

\239       3(239)3  ^  5(239)^       7(239)'  ^/  ^ 

Multiplying  both  members  by  4,  we  find, 


tan 
hence 


arc  45°  = 


180' 


3.141592653. 


the  diflference,    4a  —  45°  =  A  —  B  =  b,    will  have  for  its  tangent, 

,  ,        „  tan  A  —  tan  B  1 

tan  h  =  tan  {A  —  B)  —  ,    ,    . — -7— — 5  =  TnTTT? 
^  '         1  +  tan  A  tan  B        239 ' 


hence,  four  times  the  are  whose  tangent  is    — ,    exceeds  the  are  of  45* 


by  an  are  whose  tangent  is 


239' 


SECTION      YII. 

TRANSCENDENTAL      CURVES CURVATURE RADIUS       OF 

CURVATURE INVOLUTES    AND   EVOLUTES. 

Classification    of   Curves. 

101.  Curves  may  he  divided  into  two  general  classes  : 
Ist.     Those   -whose   equations   are   purely  algebraic;    and 
2dly.  Those  whose  equations  involve  transcendental  quan- 
tities. 

Those  of  the  first  class,  are  called  Algebraic  curves,  and 
those  of  the  second,   Transcendental  curves. 

The  properties  of  the  Algebraic  curves  have  been  already 
ex.aniined;  it  therefore  only  remams  to  explain  the  proper- 
tics  of  the  Transcendental  curves. 

Logarithmic    Curve. 

102.  A  logai'ithmic  curve,  is  a  curve  in  which  one 
of  the  90-ordinates,  of  any  point,  is  the  logarithm  of  the 
other.  The  co-ordinate  axis  to  which  the  lines  denotiiiir 
The  lorarithras  are  parallel,  is  called  the  axis  of  logarithms^ 
and  the  other,  the  axis  of  numhers. 

If  we  suppose  Y  to  be  the  axis  of  logarithms,  then  X 
will  be  the  axis  of  mmibers,  and  the  e(]untion  of  the  curve 
will  be, 

y  z=z  log  X. 


140 


DIFFER  E:NTIAL      CALlULCS. 


[sec.  vil 


General    Properties. 

103.  Let  A.  be  the  origin  of  a  system  of  rectangular 
co-ordinates,  X  the  axis  of  numbers,  and  T  the  axis  of 
logarithms. 

If  we  designate  the  base 
of  a  system  of  logarithms 
by  ff,  we  shall  have,* 

a"  =  X, 

in    which    y    is    the    logar 

rithm  of  x. 

If  we   change   the  value 

of  the   base   a,   to   a\   we   shall  have, 

y 
a'    —  X, 

in  which  y  is  the  logarithm  of  a*,  to  the  base  a\  It  is 
plain,  that  the  same  value  of  ic,  in  the  two  equations,  will 
give  different  values  of  y,  and  hence  :  £Jach  si/stem  of 
logarithms  will  gire  n  fJifferent  logarithmic  curve. 

If  we  make  y  =  0,  Ave  shall  have,t  x  =  1;  and  since 
thi"^  relation  is  independent  of  the  ba^e  of  the  system  of 
logarithms,  it  follows,  that :  Ecery  logarithmic  curve  will 
intersect  the  axis  of  ^mmhers  at  a  distance  from  the  origin 
equal  to   1. 

This   abscissa  is   denoted   by  the  line  AE. 

We  may  find  points  of  the  cui^e  from  the  general 
equation, 

a*  =:  a?, 


*  Bourdon,  Art.  a^T.     University,  Art.  l«:?. 
f  Bourdon,  Art.  235.     Universiiy,  Art.  1§6. 


SEC.  VII.]  TRAJfSCENDENTAL      CUKVES.  141 

even  without  the  aid  of  a  table  of  logarithms.  For,  if 
we   make, 

1                   3  1        - 

y  =  0,       y  =  2'       2/ =  2'  ^  "^  I'         ' 

we   shall   find,  for  the   corresponding  values   of  jc, 

X  =  I,        X  zzzy/a^       X  =  a-y/tt,        X  =  l/a,    &c. 

If  we  make  a  =  10,  the  curve  will  correspond  to  the 
common  system  of  logarithms  ;  and  if  we  suppose 
a  =  2.7182818...,  to  the  Naperian  system.  Both  curves 
pass  through   the   point   JEJ. 

Base  >  1. 

104.  If  we  suppose  the  base  of  the  system  of  loga- 
rithms to  be  greater  than  1,  the  logarithms  of  all  numbers 
less  than  1  will  be  negative;*  therefore,  the  values  of  y, 
corresponding  to  all  abiseissas  between  the  limits  of  a;  =  0, 
and  X  -=  AE  =  1,  will  be  negative;  honce,  these  ordi- 
nates  are  laid  off  below  the  axis  of  JT.  AMien  x  ■=  0, 
y  =  —  00.  Therefore,  when  tlie  base  is  greater  than  1,  the 
corresponding  curve  is  QPEK' .  The  curve  cannot  extend 
to  the  left  of  the  axis  of  I",  since  negative  numbers  have 
no  real  logarithms.f 

Base  <  1. 

^  105.  If  the  base  of  the  system  is  less  than  1,  the  log- 
aritlims  of  all  numbers  greater  than  1  are  negative;  and 
of  all  numbers  less  than  1,  positive.  Under  this  supposi- 
tion, the  curve  assumes  the   position   Q'I'EK.      The  parts 


*  Bourdon,   Art.  ^J>5.     University,  Art.   1  gC 
\  Bourdon,  Art,  33(3.     University,  Art.   1S6 


142  DIFFEKEXTIAL      CALCULUS.  [SEC.  VU. 

of  the  curves  EPQ^  EP  Q\  are  concave  towards  the  aids 
of  abscissas;  the  parts  EK^  EK\  are  convex;  and  both 
curves,  throughout  their  whole  extent,  are  convex  towards 
the   axis  of   Y. 

Asymptote. 

106.  Let   us  resume  the   equation  of  the   curve, 

y  —  logic. 

If  we  denote  the  modulus  of  a  system  of  logarithms  by 
3f,    and   differentiate,   we  have    (Art.  88), 

dx^^  dy        M 

dy  =  -Jf;        or,        ^^  =  -• 

But,  -^    denotes  the   tans^ent    of  the    an  He   which    the 
dx 

tangent  line  makes  with  the  axis  of  abscissas;  hence,  the 
tangent  will  be  parallel  to  the  axis  of  abscissas  when 
a;  =  00,    and   perpendicular   to   it,    when  x  =  0. 

But,  when  x  =  0,  y  =  —  cc;  hence,  the  axis  of  ordinates 
is  an  asymptote  to  the  curve.  The  tangent  which  is  par- 
allel to  the  axis  of  -Z",  is  not  an  asymptote;  for,  when 
jc  =  00 ,   we   also   have,  y  =  co    (Art.  50). 

Sub-tangent. 

107.  The  most  remarkable  property  of  this  cui-ve,  is  the 
value  of  its  sub-tangent  T'Ii\  estimated  on  the  axis  of 
logarithms.  We  have  found,  for  the  sub-tangent,  on  the 
axis  of  X  (Art.  45), 

-^ = %"■■ 


BBC.  VII.]  TRANSCENDENTAL      CUKVKS.  143 

and  by  simply  changing  the  axis,  we  have, 

T'R'  =  -j^x  =  M   (Art.  106);        hence, 

ax  ^  ' 

The  sub-tangent^  taken  on  the  axis  of  logarithms,  is  equal 
to  the  modulus  of  the  s'gstem  from  which  the  curve  is 
constructed.  In  the  Napeiian  system,  M  =  \\  hence,  the 
sub-tangent  is  equal  to  1,  equal  to  AE.  In  the  common 
system,  it  is  denoted  by  the  number,  .434284482 . . . 


108.  If  a  circle  KFG  be  rolled  along  a  straight  line, 
AL,  any  point  of  the  circumference,  as  P,  will  describe 
a  curve,  which  is  called  a  cycloid.  The  circle  NPG  is 
called  the  generating  circle,  and  1\  the  generating  point. 

Since  each  succeeding  revolution  of  the  generating  circle 
will  describe  an  equal  curve,  it  will  only  be  necessary  to 
examine  the  properties  of  the  curve  APJ^L,  described  in 
one  revolution.  We  shall,  therefore,  refer  only  to  this 
part,  when  speaking  of  the  cycloid. 

If  we  suppose  the  point  P  to  be  on  the  line  AL,  at  A, 
it  will  be  found  at  some  point,  as  X,  after  all  the  points 
of  the  circumference  shall  have  been  brought  in  contact 
with  the  line  AL.  The  line  AL  will  be  equal  to  the 
circumference   of   the   generatmg    circle,   and   is   called   the 


vu 


DIFFERENTIAL      CALCULUS. 


[sec.  vn. 


brfse  of  the  cycloid.  The  line  -Z?J/',  drawn  perpendicular 
to  the  base,  at  tlie  middle  point,  is  called  the  axis  of  the 
cycloid^  and  is  equal  to  the  diameter  of  the  generating 
circle. 

Transcendental    Equation    of   the    Cycloid. 

109.  Let  CN  be  the  radius  of  the  generating  circle. 
Assume  any  point,  as  A,  for  the  origin  of  co-ordinates. 
Let  us  suppose  that  when  the  generating  point  has  de- 
scribed any  arc  of  the  cycloid,  as  ^-P,  that  the  point  in 
which  the  circle  touches  the  base  has  reached  the  point  N, 


Through  N^  drnw  the  diameter  KG^  of  the  generating 
circle  :  it  will  be  perpendicular  to  the  base  AL.  Through 
P.  draw  PR  perpendicular  to  the  base,  and  PQ  parallel 
to  it.  Then,  PP  —  KQ  will  be  the  versed  sine,  and  PQ 
the   sine   of  the   arc   NP  to   the   radius    CN".      Pat, 

CR  =  r,        AP  =  X,        PP  =  JSTQ  =  y, 

we   shall  then   have, 


PQ  =  -v/2ry-2/%    ^  =A]Sr-PN'=  arc  NP-PQ-, 
hence,   the   transcendental   equation   is, 


ver-sin~^y  —  ^/2ry  —  y"^. 


8K0.  Vn.]  TEANSCENDENTAL      CUKVES.  145 


Differential    Equation. 

110.  The  properties  of  the  cycloid  are  most  easily 
deduced  from  its  differential  equation.  This  is  found  by 
differentiating  both  members  of  the  transcendental  equation. 
We  have   (Art.  97), 

Tcly 
<?(ver-sin-^y)  =  _;    and 

V2ry  -  2/2 

d{-  V^  -  2/0  =   -  ^^^=^=^;    hence, 
V  2^2/  —  2/2 

^^         r^___n7y-j^.     or,      dx  =        ^^^       ■ 


\/2ry  -  2/^        '/2ri/  -  y^'         '  ^y  ^  yi' 

which   is  the   differential   equation   of  the   cycloid. 

Sub-Tangent,    Tangent,    Sub-Normal,    Normal. 

111.    If  we  substitute  in   the   general   equations   of  Arts. 

45,  46,  47,   and  4§,   the  value  of  ^^    found     in     the    dif 

dx 

ferential  equation  of  the  cycloid,  we    shall   obtain  the  values 

of  the  sub-tangent,  tangent,  normal,  and  sub-normal. 

TR  =  --=A =  sub-tangent ; 


TP  = 


RN  =  y^y  —  2/2  _  sub-normal. 


V2ry  -  y2 

y^lry 

V'^W  -  2/^ 

'v/2ry 

146 


DIFFERENTIAL      CALCULUS.  [SEC.  VH. 


These  values  are  easily  constructed,  from  their  conneo- 
tion  with  the  parts  of  the  generating  circle. 

The  sub-normal  BN^  for  example,  is  equal  to  PQ  of 
the  generating  circle,  since  each  is  equal  to  -y/lry  —  y^j 
hence,  the  cormal  PiV,  and  the  diameter  GN",  intersect 
the  base  of  the  cycloid  at  the  same  point.  Now,  since 
the  tangent  to  the  cycloid  at  the  point  P  must  be  per- 
pendicular to  the  normal,  it  will  coincide  with  the  chord 
PG  of  the  generating  circle. 

If,  therefore,  it  be  required  to  draw  a  normal,  or  a  tan- 
gent, to  the  cycloid,  at  any  point,  as  P,  draw  any  line, 
as  ng^  perpendicular  to  the  base  AL^  and  make  it  equal 
to  the  diameter  of  the  generating  circle.  On  ng^  describe 
a  semi-circumference,  and  through  P  draw  a  parallel  to 
the  base  of  the  cycloid.  Through  p,  where  the  parallel 
cuts  the  semi-circumference,  draw  the  supplementary  chords 
jt>w,  pg^  and  then  draw  through  P  the  parallels  PN^ 
PG\  and  PN  will  be  a  normal,  and  PG  a  tangent  to 
the  cycloid  at  the  point  P. 


Position   of  Tangent. 
112.    The  differential  equation  of  the  curve, 


dx 


v^y  -  y^ 


SEC.  VII.]  TRANSCENDENTAL      CURVES.  147 

may  be  put  under  the  form, 

^  ^   V2ry  -  y^   _     /5r  __  ^ 
dx  y  ~~  V  y 

If  we  make  y  =  0,  we  shall  have, 

dx  ' 

and  if  we  make  y  =  2r,  we  shall  have, 

dx 

hence,  the  tangent  lines  drawn  to  the  cycloid  at  the  pointa 
where  the  curve  meets  the  base,  are  perpendicular  to  tlie 
base ;  and  the  tangent  drawn  through  the  extremity  of  the 
greatest  ordinate,  is  parallel  to  the  base. 

Curve    Concave. 
113.    If  we  diflerentiate  the  equation, 

dx  =  ——= _, 

V2ry  —  2/2 

regarding   dx  as  constant,  we  obtain, 

V2ry  -  y2 
or,  by  reducing  and  dividing  by  y, 

0  =  (2ry  —  y^)d^y  +  rdy\ 

whence  we  obtain, 

rdy^      ^ 

^^-         2^-y^' 

and  hence,  the  curve  is  concave  towards  the  axis  of 
abscissas    (Art.  ya). 


us 


DIFFERENTIAL      CALCULUS.  [SEC.  VU, 


Area    of  the    Cycloid. 

114.  The  area  of  the  cycloid  may  be  found  in  a  very 
simple  manner,  by  constructing  the  rectangle  AFBM^  and 
considering   the   portion   AFB, 

If  we  regard  F  as 
an  origin  of  co-ordinates, 
FB  as  a  line  of  ab- 
scissas, and  take  any 
ordinate,   as, 

KH 


z  =  2r 


y> 


we  shall  have, 
Rut, 


d{AHKF)  =  zdx. 


zdx  =  — — _yM^  _  ^y^2i'y  —  y2; 
V2ry  —  y2 


whence,  AHKF  =fdy^/^—  y^  +  (7. 

But  this  integral  expresses  the  area  of  the  segment  of 
a  circle,  whose  radius  is  r,  and  versed-siiie  y  (Art.  99), 
that  is,  of  the  segment  MIGE.  If  now,  we  estimate  the 
area  of  the  segment  from  J/,  where  y  =  0,  and  the  area 
AFXH  from  AF^  in  which  case  the  area  AFKH  —  0, 
for  2/  =   0,  we   shall  have, 

AFKR  =  MIGE; 

and  taking    the    integral    between    the  limits  y  =  0   and 
y  =  2r,   we   have, 

AFB  =  semi-circle  MIGB, 

and   consequently, 

area  AIIBJSI  =  AFBM  -  MIGB. 


SEC.  VII.]  T  R  A  N  S  C  E  N  D  E  X  TA  L      CURVES.  319 

But  the  base  of  the  rectangle  AFBM  is  equal  to  the 
semi-circumference  of  the  generalhig  circle,  and  the  alti- 
tude is  equal  to  the  diameter ;  hence,  its  area  is  equal 
to  four  times  the  area  of  the  semi-circle  3IIGB ;  there- 
fore, 

area  AHBM  =  33fIGB ;  hence. 

The  area  AHBL  is  equal  to  three  times  the  area  of 
the  generating  circle. 

Surface  described  by  tho  Cycloid. 

115.    To  find  the   surface    described    by   the    arc   of   a 
cycloid  when   revolved   about   its  base. 
The  differential  equation  of  the  cycloid  is, 


6x=    y^y 


Substituting  this  value  of  dx  in  the  differential  equation  of 
the   surface   (Art.  62),   it  becomes, 

Applying  Formula  (E),   (Art.  170),  we  have, 

L      3  3   y  ^2rg  -  yd 

But, 

23 


150  DIFFERENTIAL      CALCULUS.  [sEC.  VH. 

heiKie, 

8  =   2;rV2r[-  ?2/^v^  -y'-  lr{2r  -  2/)^]  4-  C. 

If  we  estimate  the  sui-fiice  from  the  plane  passing  through 
the  centre,  we  have  C  =  0,  since  at  this  point  5  =  0, 
and  7/  z=  2r.  If  vre  then  integrate  between  the  limits 
y  =  2r,     and    y  =  0,     we  have, 

s  =  -  surface  =    —   — cr-;    hence, 

2  o 

s  =  surface  =   —  — cr"'*, 
3        ' 

that  is,  the  surfiice  described  by  the  cycloid,  when  it  is 
revolved  around  the  base,  is  equal  to  64  thirds  of  the 
generating   cii'cle. 

The  minus  sign  should  appear  before  the  integral,  since 
the  surface  is  a  decreasing  function  of  the  variable  y 
(Art.  19). 

Volume  generated  by  the  area  of  the  Cycloid. 

116.  If  n  cycloid  be  revolved  about  its  base,  it  is  re- 
quired to  find  the  measure  of  the  volume  which  the  area 
^11  generate. 

The  differential  equation  of  the  cycloid  is, 


-v/27^-  y2 

If  we  denote  the  volume  by  T",  we  have   (^Art.  60), 

'^fifcly 


dV  = 


V^ri/  —  y* 


SEC.  Vn.]  TEANSCENDENTAL      CURVES 


151 


If  we  apply  Formula  (E)  (Art.  170),  we  shall  find,  aftoT 
three  reductions,  that  the  integral  will  depend  on  that  of 

dy 
^2ry  -  2/2 

But  the  integral  of  this  expression  is  the  arc  whose  vereed 
sine  is  -  (Art.  99).  Making  the  substitutions  and  reduo- 
tions,  we  find  the  volume  equal  to  five-eighths  of  the 
circumscribing   cylinder. 

Spirals. 

117.  A  Spiral^  or  Polar  Line,  is  a  curve  described  by 
a  point  which  moves  along  a  right  line,  according  to  any 
law  whatever,  the  line  having  at  the  same  time  a  uniform 
angular  motion. 

Let  AJBG  be  a  straight 
line  which  is  to  be  turned 
uniformly  around  the  point 
A.  When  the  motion  of 
the  line  begins,  let  us  sup- 
pose a  point  to  move  from 
A  along  the  line,  in  the 
direction  AB  C.  When 
the  line  takes  the  posi- 
tion ADE,  the  point  will 

have  moved  along  it,  to  some  point,  as  Z>,  and  will  h.nvc 
described  the  arc  AaD,  of  the  spiral.  When  the  line 
takes  the  position  AD'E\  the  point  will  have  described 
the  curve  AaBD\  and  when  the  line  shall  have  complet- 
ed an  entire  revolution,  the  point  will  have  described  the 
curve  AaBD'B. 

If  the  revolutions  of  the  radius-vector  be  continued,  the 


152  DIFFEnENTIAL      CALCULUS.  [sEC.  VII. 

generating  point  will  describe  an  indefinite  spiral.  The 
point  A,  about  which  the  right  line  revolves,  is  called 
the  pole ;  the  distances  AD,  AD\  AJS,  are  called  ra- 
dius-vectors or  radii-vectores  ;  and  the  parts  AaDD'B, 
BFF' C,   described    in   each   revolution,   are   called    spires. 

If,  with  the  pole  as  a  centre,  and  AB,  the  distance 
passed  over  by  the  generating  point  in  the  direction  of 
the  radius-vector,  during  the  first  revolution,  as  a  radius, 
we  describe  the  circumference  BEE\  the  angular  motion 
of  the  radius-vector  about  the  pole  A,  may  be  measured 
by  the  arcs  of  this  circle,  estimated  from  B. 

If  we  designate  the  radius-vector  by  u,  and  the  meas- 
nring  arc,  estimated  from  J5,  by  f,  the  relation  between 
u  and  t,  may  be  expressed  by  the  equation, 

u  =  f[t),  or  u  =  ar, 

in   which   7i    depends   on   the   law  according  to   which   the 

generating  point  moves  along  the  radius-vector,  and  a  on 

the  relation  which  exists  between  a  giv€?i  value  of  w,  and 
the  corresponding  value  of  t. 

General  Properties. 

lis.  When  n  is  positive,  the  spirals  represented  by 
the  equation, 

u  =  at^y 

will  pass  through  the  pole  A.  For,  if  we  make  ^  =  0, 
we  shall  have,   u  =  0. 

But  if  71  is  negative,  the  equation  will  become, 

a 
or,        «  =  -, 


SEC.  VII.]  TRANSCENDENTAL      CUPwVES,  153 

from  which  we  shall  have, 

for,  ^  =  0,  w  =   00, 

and  for,  ^=00,  w  =  0 ; 

hence,  in  this  class  of  spirals,  the  first  position  of  the 
generating  point  is  at  an  infinite  distance  from  the  pole: 
the  point  will  then  approach  the  pole  as  the  radius-vector 
revolves,  and  will  only  reach  it  after  an  infinite  number 
of  revolutions. 

Spiral  of  Archimedes. 

119.    If  we  make    t*  =  1,    the  equation  of  the  spiral 

becomes, 

u  =z  at. 

If  we  designate  two  different  radii-vector es  by  u'  and 
m",  and  the  corresponding  arcs  by  t'  and  t'\  we  shall  have, 

u'  =  at\        and        w"  =  at", 

and  consequently, 

w'    :    u"    :  :    t'    :    t" ;    that  is, 

The  radii-vectores  are  proportional  to  the  measuring 
arcs,  estimated  from  the  initial  point. 

This  spiral  is   called  the   spiral   of  Archimedes. 

If  we  denote  by  1,  the  distance  which  the  generating 
point  moves  along  the  radius-vector,  during  one  revolu- 
tion, the   equation. 


will  become. 


1  =  at;        or,         1  x  -  =  f . 
'  a 


154  DIFFEKEXTIAL      CALCULUS.  [SEC.  TTI. 

But    since     t    is    the    circumference    of   a    circle    whose 
radius  is    1,    we   shall    have, 

-  =  2ty        and  consequently,        a  =  —  • 


Parabolic  SpiraL 

120.  If  we  make  n  =  -,  and  a  =  v^>  w®  have, 
for  the  general  equation, 

u  =  v^  X  t^ ',        or,        u^  =  2pt, 

which  is  the  equation  of  the  parabolic  spiral. 

If  ^  =  0,  w  =  0 ;  hence,  this  spiral  passes  through 
the  pole. 

Hyperbolic  Sj^raL 

121,  If  we  make  w  =  —  1,  the  general  equation  of 
ipirals  becomes, 

u  =  at-^;        or,        ut  =  a. 

This  spiral  is  called  the  hyperbolic  spiral^  because  of 
the  analogy  which  its  equation  bears  to  that  of  the  hy- 
perbola,  when  referred  to  its  asymptotes. 

If,  in  this  equation,  we  make,  successively, 

<  =  1,        «  =  2'        *  =  3'        ^  ^  V    ^°" 

we  shall  have  the  corresponding  values, 

u  =  a,      u  =  2a,     w  =  3a,     u  =  4a,    &c. 


SBC.  Vn.]  TBANSCENDENTAL      CUEVES, 


155 


Logarithmic  Spiral. 

122.  Since  the  relation  between  u  and  t  is  entirely 
arbitrary,  we  may,  if  we  please,  make, 

t  —  log  w. 

The  spiral  described  by  the  extremity  of  the  radius- 
vector,  xmder  this  supposition,  is  called  the  logarithmic 
spiral. 

Direction  of  the  measuring  arc. 

123.  The  arc,  which  measures  the  angular  motion  of 
the  radius-vector,  has  been  estimated  from  right  to  left, 
and  the  value  of  t  regarded  as  positive.  If  we  revolve 
the  radius-vector  in  a  contrary  direction,  the  measuring 
arc  will  be  estimated  from  left  to  right,  the  sign  of  t 
will  be  changed  to  negative,  and  a  similar  spiral  will  bo 
described. 

Sub-tangent  in  Polar  Curves. 

124.  The  Sub-tangent,  in  spirals,  or  in  any  curvCj 
referred  to  polar  co-ordinates, 
18  the  projection  of  the  tangent 
on  a  line  dra\\Ti  through  the 
pole,  and  perpendicular  to  the 
padius-vector  passing  through 
the   poiut   of  contact. 

Let  A  be  the  pole,  AR  =  1, 
the  radius  of  the  measuring 
arc,  P  any  point  of  the  curve, 
TP    a  tangent  at  P,  and  Al\ 


156  DIFFEPvENTlAL      CALCULUS.  [SEC.  VH. 

perpendicular  to  AP,  the  sub-tangent.  Let  AP'  be  a 
radius-vector,  consecutive  with  AJP,  and  P§,  an  arc  de- 
scribed from  the  centre  A. 

Then,  JViV"'  =  dt,  and  QP  —  duy  and,  since  JPQ  is 
parallel  to  iViV',  we  have,  P§  =  itdL  But  the  arc 
PQ  coincides  with  its  chord  (Art.  43),  and  since  Q  ia 
a  right  angle,  the  triangles  PQP'  and  7L1P  are  similar; 
hence, 

AT  :  AP  ::  PQ  :   QP' ;    therefore, 

Sub-tangent     AT  :    u     :  :  udt  :    du. 

Whence,         Sub-tangent  AT  =  "^  =  -t--^\ 

du         n 

125.     In   the   spiral   of  Archimedes,   we  have, 

1 
n  =  1,        and        a  =  ~; 

hence,  AT  =  --^ 

If  we  make  t  =  2^,  circumference  of  the  measuring 
circle,  we   shall  have, 

AT  =  2^,  circumference   of  the  measuring  circle. 
After  m  revolutions,  we   shall  have 

t  =  2mie, 
and   consequently, 

AT  =  2mV  =  m.27n'!r-j  that  is, 
The  8iib-tanr/eJit,  after  in  revQlutions^  is  equal  to  m  times 


SEC.  VII.]  TRANSCENDENTAL      CUEVES.  157 

the  circumference  of  the  circle  whose  radius  is  the  radiuS' 
vector.      This  property  was   discovered  by  Archimedes. 

126.  In  the  hyperbolic  spiral,  ?z  =  —  1,  and  the  value 
of  the  sub-tangent  becomes 

AT  =z  —  a ;   that  is, 

The  sub-tangent  is  constant  in  the  hyperbolic  spiral. 

Angle  of  the  Tangent  and  Radius-Vector. 

127.  We  see  that, 

AT  _  icdt 
AP  ~  du  ' 

denotes   the  tangent  of  the   angle  which   the   tangent  line 
makes   with   the   radius-vector. 

In  the  logarithmic   spiral,  of  which  the   equation  is 


we  have, 


hence,  -j-p  =  -y-  =  M;    that  is, 


In  the  logarithmic  sjyiral,  the  angle  formed  by  the 
tangent  and  the  radius-vector  passing  through  the  point 
of  contact^  is  constant ;  and  the  tangent  of  the  angle 
is   equal  to  the  modulus   of  the   system  of  logarithms. 

If  t  is  the  Naperian  logaritlim  of  w,  JLT  is  1  (Art.  89), 
and   tlie   angle  will   be   equal   to   45°. 


t 

= 

log  u, 

dt 

= 

du 
—M; 

AT 
AP 

= 

udt 

da  ~~ 

158  DIFFERENTIAL      CALCULUS.  ] SEC.  VII, 

Value    of   the    TangenL 

128.  The  value  of  the  tangent,  in  a  curve  referred  to 
polar  co-ordinates,  is, 

Dififerential   of  the   Arc. 

129.  To  find  the  differential  of  the  arc,  which  we  de- 
note by  2,  we  have, 

or,  by  substituting  for   PP\    QP\   and  P§,  their  Values, 
when  P  and  P'  are  consecutive,  we  have, 


Differential    of  the    Area. 

130.  The  differential  of  the  area  ADP,  when  referred 
to  polar  co-ordinates,  is  not  an  elementary  rectangle,  as 
when  referred  to  rectangular  axes,  but  is  the  elementary 
sector  APP'.  The  area  of  this  triangle  is  equal  to 
AP  X  PQ  ^     j^  ^^  denote  the  differential  by  ds,  we  have, 

AP'x  QP        (u  -f  du)udt 
^^  =   2 =  2  * 

or,   omitting    the    infinitely  small    quantity   of   the    second 

order,,  ududt    (Art.  20), 

u^dt 
ds  =  -^, 


SEC.  VII.]  TKANSCENDEXTAL      CURVES.  159 

which  is  the  differential  of  the  area  of  any  segment  of  a 
polar  line.  • 

Areas  of   Spirals. 

131.  If  we  denote  by  5,  the  area  described  by  the  ra- 
dius-vector, we  have  (Art.  130), 

ds  =  — ; 

and  placing  for  u  its  value,   at""  (Art.  11  t), 

ds  z=  — - —  ,         and         s  = +  C. 

If  ?*  is  positive,   C  =  0,  since  the  area  is  0,  when  t  =  0. 
After  one  revolution  of  the  radius-vector,  t  =  2if,  and  we 

have, 

_   g"(2r)^'»  +  ^ 

^   ""   ~4fi  +  2     ' 
which  is  the  area  included  within   the  first  spire. 

132.  In  the  spiral  of   Archimedes,   (Art.  119), 

1 
a  =  -—,         and        n  =  1  : 

hence,  for  this  spiral  we   have, 

^   -    2172' 

which  becomes   - ,   after   one   revolution   of  the   radius-veo- 
3 

If 
tor ;    the    unit   of  the   number    -  ,    being    a   square   whose 

o 

side  is  1.  Hence,  the  area  included  by  the  first  sjvre^  is 
equal  to  one-third  of  the  area  of  the  circle  whose  radius 
is  the  radius-vector^  after  the  first  revolution. 


ICO  D  I  F  F  E  i:  E  N  T  I  A  L      CALCULUS.  [SEC.  VH. 

Ill  the  second  revolution,  the  radius-vector  describes  a 
second  time,  the  area  described  in  the  first  revohition ; 
and  in  any  succeeding  revohition,  it  will  pass  over,  or  re- 
describe,  all  the  area  before  generated.  Hence,  to  find 
the  area,  at  the  end  of  the  mth  revolution,  we  must  in- 
tegrate between  the  limits, 

t  =  {m  —  1)27^,         and         t  =  m.2'!f, 

which  gives,' 

If  it  be  required  to  find  the  area  between  any  two  spires, 
as  between  the  ?nth  and  the  (m  +  l)th,  we  have  for  the 
whole  area  to  the  (m  +  ])th  spire, 

3  .  ' 

and  subtracting  the  area  to  the   mth.   spire,  gives, 

(m  4-  1)^  —  2m3  +  hn  —  If 
s= ^^ -'Tf  =  2m^, 

o 

for  the  area  between  the  mth  and  {m  +  l)th  spires. 

If  we  make  m  =  I,  we  shall  have  the  area  between 
the  first  and  second  spires  equal  to  2':f ;  hence,  the  area 
between  the  mth  and  {m  -h  l)th  spires,  is  equal  to 
m    times   the  area  between   the  first  and  second. 

133.     In  the  hyperbolic  spiral,   ?i  =  —  1,  and.  we  have, 

ds  =  -^^^^        a°<i        5  =   -  -• 
The   area    s    will  be  infinite,   when    ^  =  0,   but  we   can 


6EC.  VIl]  CFRVATURE.  161 

find  the  area  included  "between  any  two  radii-vectores  h 
and  c,  by  integrating  between  the  hmits  t  =  h  and  t  =  c^ 
which  will  give, 

134.     In    the    logarithmic    spiral,    t  =  lu  \    hence, 

dt  =  — 

u  '  ,       li^dt        udu 

and,      —  =  —; 

Pudu        y^   ,    ^ 
hence,  s  =z  J  -^  z=.  -  -\-  C ; 

and  by  considering  the  area  5  =  0,  when  w  =  0,  we 
have    (7=0,    and 


C  URVAT  UR  E. 

135.  The  Cuevatuee  of  a  plane  curve,  at  any  point, 
is  the  departure  from  the  tangent  drawn  to  the  curve  ai 
that  point.  This  departure  is  measured  by  the  distance 
which  a  point,  moving  on  the  curve,  departs  from  the 
tangent  in  passing  over  a  unit  of  length,  denoted  by  the 
differential  of  the  arc.  In  the  same  circle,  or  in  equal 
circles,  the  departure  from  a  tangent,  at  any  point,  is 
always  the  same ;  hence,  the  curvature  of  a  circle,  at  all 
points,  is  constant. 

Curvature  of   a  circle  is  inversely  as  the  radius. 

136.     Let  G  and   C  be  the  centres  of  two  unequal  cir 
cles,  having  a  common  tangent  at  P.      If  we  suppose  the 


162 


DIFFERENTIAL      CALCULUS, 


[sec.  Vll. 


arcs  to  be  the  independent  variables,  we  can  denote  the 
differential  of  one  arc  by  Ph^  and 
the  differential  of  the  other,  by 
an  equal  arc  Pa.  Then,  having 
drawn  hH  and  aD^  and  the  sines, 
bh\  aa\  and  recollecting  that  each 
arc  is  equal  to  its  corresponding 
chord,  (Art.  43),  we  liave,  by  de- 
noting the   radii  by  r  and  r',* 

W  =:  2r.Pb',         and        W  =  2r'.Pa'; 

since  the  arcs  are  equal,  and  PJ'  =  db,    and  Pa'  =  ca^ 

2r.db  =  2r'.  ca  ;     hence, 


db 


ac 


that  is, 


The  curvature  of  a  circle  varies  inversely  as  its  radiits  / 
hence,  the  reciprocal  of  the  radius  of  a  circle  may  be 
assumed  as  the  measure  of  its  curvature. 

Orders  of  Contact. 

137.  If  two  plane  curves  have  one  point  in  common, 
there  is  one  set  of  co-ordinates  (which  may  be  denoted 
by  x",  y"),  that  will  satisfy  the  equations  of  both  curves. 
If  the  curves  have  a  second  point  in  common,  consecutive 
with  the  first,  they  will  have  a  common  tangent,  at  the 
common  point,  and  the  first  differential  coefficients  will  also 
be  equal  (Art.  43)  ;  this  is  called,  a  contact  of  the  first  order  ^ 
If  the  curves  have  a  third  .  point  in  common,  consecutive 
with  the  second,  the  second  differential  coefficients  will  be 


*  Legendre,  Bk.  IV.  P.  23. 


SEC.  VII.]  CUEVATUEE,  ]63 

equal   (Art.  73)  ;   this  is   called,   a  contact   of  the    secotid 
order. 

Generally,  two  curves  have  a  contact  of  the  nth  order, 
when  they  have  a  common  point,  and  the  first  n  succes- 
sive differential  coefficients  of  the  common  ordinate,  equaJ 
to  each  other. 

Oscillatory  Curves. 

13§.  An  Osculatkix,  is  a  curve  which  has  a  highet 
order  of  contact  with  a  given  curve,  at  a  given  point, 
than  any  other  curve  of  the  same  kind.  The  osculatory 
circle  is  by  far  the  most  important  of  all  the  osculatrices ; 
for  it  is  this  circle  which  measures  the  curvature  of  all 
plane  curves. 

Oscillatory  Circle. 

139.  The  general  equation  of  a  circle,  referred  to  rect^ 
angular  co-ordinates   (Bk.  II.,  Art.  »^),  is, 

(x-ay+  {y-  ^y  =  1^    .     .     .     (1.) 

in  which   a  and  ^   are  ike  co-ordinates  of  the  centre,  and 
X  and   2/  the  co-ordinates  of  any  point  of  the  curve. 

If  we  regard  a,  /3,  and  Jl,  as  constants,  and  difll.i<  n- 
tiate  the  equation  twice,  and  then  lind  the  differential 
coefficients  of  the  first  and  second  order,  we  have, 


dy   _        X  —  a 
dx   ~  "  y  -  ^ 


(2.) 


1+^ 
and,  d^y  dx^     ....     (3.) 

dx^  ~  y  —  13 


164  DIFFEKENTIAL      CALCULUS.  [s EC.  VII. 

In  Equation  ( 1 )  there  are  three  arbitrary  constants,  a, 
/3,  and  li  ;  and  values  may  Le  assigned  to  these,  at  pleas- 
ure, so  as  to  cause  the  circle  to  fullfil  three  conditions, 
and  three  only. 

If  we  have  any  plane  curve  whose  equation  is  of  the 
form, 

V  =  A^) ; 

and  find,  from  this  equation,  the  first  and  second  differ- 
ential coefficients,  for  any  point  whose  co-ordinates  are 
25"?  y'\  ^e  may  then  attribute  such  values  to  a,  /8,  and 
7?,   as   shall  make, 

X  =  x",        y  =  y"',       also, 

^  _  ^       and      ^'y  _  ^y" 

dx  ~  dx"'  dx^  ~  dx"^' 

As  no  further  general  relations  can  be  established  be- 
tween the  differential  coefficients  of  the  circle  and  curve, 
this  circle  will  be  oscillatory  to  the  curve  at  the  point 
whose  co-ordinates  are  ic",  y"  (Art.  138).  Since  the 
co-ordinates  of  a  point,  and  the  differential  coefficients  of 
the  first  and  second  order,  determine  three  consecutive 
points  (Art.  73),  it  follows  that,  the  osculatory  circle  passes 
through  three  consecutive  points  of  the  curve,  at  the  point 
of  osculation. 

Limit  of   the   Orders   of   Contact. 

140.  It  is  seen  that  the  highest  order  of  contact  which 
a  circle  can  have  with  any  curve,  is  denoted  by  the  num- 
ber of  arbitrary  constants  which  enters  into  its  equation, 
less    1;    and    the    same    is   true    for   any   other   osculatrix. 


SEC.  VII.] 


CURVATURE. 


165 


Although  it  is  impossible  to  assign  a  higher  order  of  con- 
tact, to  a  circle,  than  the  second^  yet,  at  the  vertices  of 
the  transverse  and  conjugate  axes  of  the  conic  sections,  the 
conditions  which  make  the  circle  osculatory,  also  mal:e  the 
third  differential  coefficient  zero,  and  hence  give  a  contact 
of  the  third  order.  In  general,  when  the  order  of  contact 
is  evew,  and  the  curve  symmetrical  with  the  normal  at  the 
point  of  osculation,  tlie  conditions  imposed  will  give  a  con- 
tact of  the  next  hifjLer  order. 


Radius   of  Ou  vature. 


141,    If  we   find   the   value   of  R  from  Equations   (1), 
(2),  and  (3),  we  have, 

{dx^  4-  dy'^Y 
dxd^y 


B  =  ± 


(4.) 


which  is  the  general  value  for  the  radius  of  the  osculatory 
circle. 

If  we  denote  the  arc  by  2,  we  have   (Art.  52), 


whence, 


dz  =    ■\/dx'^  +  c?y'; 
dz^ 


B  =   ±  - 


dxd~y 


(5.) 


Measure   of  Curvature. 

142.    The  curvature  of  a  curve,  at  any  point,  is  measured' 
by  the  curvature  of  the  os- 
culatory circle  at  that  point ; 
hence,  it  is  the  reciprocal  of 
the  radius  (Art.  136). 

If  we  assume  two  points, 

P   and    /*,    either    on    the 

24 


166  DIFFEKEXTIAL      CALCULUS.  [SEC.  VII. 

same,   or  on   dilTerent   curves,   and   find   the   radii  r   and   r' 
of  the  circles  which  are  osculatory  at  these  points,  then, 

curvature   at  P  :   curvature   at  P'  :  :    -   :   —.• 

r      r 

143.    To  find   the   radius  of  curvature,   at  any  point  of 
a  plane  curve,  whose  equation  is  of  the  form, 


y  ^fp"). 

Differentiate  the  equation  twice,  and  substitute  the  values 
of  the  first  and  second  differentials  in  Equation  (4);  the 
resulting  equation  will  indicate  the  value  of  H  for  that 
point. 

If  we  use  the  +  sign,  when  the  curve  is  convex  toward 
the  axis  of  abscissas,  and  the  —  sign  when  it  is  concave, 
the  essential  sign  of  R  will  be  positive,  w^hen  72  is  an 
increasing   function    of  x. 

Radius  of  Curvature  for  Lines  of  the  Second  Order. 

144.  The  general  equation  for  lines  of  the  second  or- 
der   (Bk.  V,  Art.  42),   is, 

2/2  =  mx  +  oix^, 
^hich   gives,  by  differentiation, 
^y  ^  (^^  +  '^nx)dx^     ^^^2  ,    ^y.  _  [V  +  (^  +  2nxy]dx^^ 

_  Inydx^  —  (m  4-  2nx)dxdy  __  [Aiiy^  —  (m  -{■  2nxY']dx'^ 
^y  —  2y^  —  ^3 


SEC.  VIT.]  CURVATUKE.  167 

!  in  the  equ 
(dx'  +  dy'^y 


Substituting  these  values  in  the  equation, 
R  ^  - 


dxcP-y       ' 

,     .                  „         \4.imx  +  nx^)  +  (w  +  InxYV 
we  obtain,  R  =  t_^ ——^ ^\ 

which  is  the  radius  of  curvature  in  lines  of  the  second 
order,  for  any  abscissa  x. 

145.  If  we  make  a;  =  0,  we  have, 

2  A  ' 

that  is,  in  lines  of  the  second  order,  the  radius  of  curva- 
ture at  the  vertex  of  the  transverse  axis  is  equal  to  half 
the  parameter  of  that  axis. 

146.  If  it  is  required  to  find  the  value  of  the  radius  of 
curvature  at  the  vertex  of  the  conjugate  axis  of  an  ellipse, 
we   make    (Bk.  V,  Art.  42), 

2i?2  B^ 

m  =  — T",        w  =   —  -rr,        and        x  =z  A, 
A  A^ 

which  gives,  after  reducing, 

i2  =  -^  ;    hence, 
Ji 

The  radius  of  curvature  at  the  vertex  of  the  conjugatt 
axis  of  an  ellipse  is  equal  to  half  the  parameter  of  that 
axis. 

14  y.  In  the  case  of  the  parabola,  in  which  n  =  0,  the 
general  value  of  the  radius  of  curvature  becomes, 

_  (m^  4-  4.mxY 
2m2         ' 


108 


DIFFERENTIAL      GALCULUS.  [SEC.  VII. 


If  we  make  a;  =  0,  -vve  shall  have  the  radius  of  curvature 


m 


at  the  vertex,   equal  to  — ,   or  one-half  the  'parameter. 

148.  If  we  compare  the  value  of  the  radius  of  curvature 
(Art.  144),  with  that  of  the  normal  line  found  in  Ait.  49, 
we  shall  have, 

^         (normal)  3 
B,  =  -5^— ^;    that   IS, 

4 

The  radius  of  cwvature,  at  any  pomt^  is  equal  to  the 
cube  of  the  normal  dimded  hy  half  the  parameter  squared; 
and  hence,  the  radii  of  curvature^  at  different  points  of 
the  same  curve,  are  to  each  other  as  the  cubes  of  th^  cor- 
responding normals ;  and  the  curvature  is  proportional  to 
the  reciprocals  of  those  cubes. 


Evolate  Curves. 

149.     Ax  EvoLUTE  curve  is  the  locus  of  the  centres  of 
all    the    circles   which   are    oscu- 
latory  to    a    given    curve.      The 
given    curve   is  called  the  Invo- 
lute. 

If  at  different  points,  P,  P', 
P",  <S:c.,  of  an  involute,  or  given 
curve,  normals,  PC,  P'C,  <fcc., 
be  drawn,  and  distances  laid  off 
on  them,  on  the  concave  side  of 

the  arc,  each  equal  to  the  radius  of  curvature  at  the 
point,  then  the  curve  drawn  through  the  extremities  (7, 
C,  C'\  <tc.,  of  these  radii  of  curvature,  is  the  evolute 
curve. 


SEC.  VII.]  INVOLUTES      AND      EVOLUTES.  169 


A  normal  to  the  Involute  is  tangent  to  the  Evolute. 

150.  Resuming  the  consideration  of  the  first  three 
equations  of  Art.  139,  and  changing  slightly  the  forms  of 
(2)  and  (3),  we  have,  I 

(x-ay+  {y  -  ^y  =  i?2  .  .  .  (1.) 
{x  -  a)dx  +  (y  -  ^)dy  =  0  .  .  .  (2.) 
dx^  +  dt/^  +  (y  -  ^)dhj  =  0      .     .     .     (3.) 

Equations  (2)  and  (3)  were  derived  from  Equation  (1), 
under  the  supposition  that  a,  /3,  and  7?,  were  arbitrary- 
constants,  and  of  such  values  as  to  cause  the  circle  to  be 
osculatory  to  a  given  curve,  at  a  given  point. 

If  now,  we  suppose  the  osculatory  circle  to  move  along 
the  involute,  continuing  osculatory  to  it,  the  five  quanti- 
ties, i?,  a,  (3,  y,  c?y,  will  all  be  functions  of  the  indepen- 
dent variable  x,  and  a  and  ^  will  be  the  co-ordinates  of 
the  evolute  curve. 

If  we  differentiate  Equations  ( 1 )  and  ( 2 )  under  this  hy- 
pothesis, we  have, 

(x  -  a)dx  +  (y  -  ^)dy  -{x-  ci)dcL  -{y-  (3)d,3  =  EdR, 

dx'  +  ^y2  +  (y  _  ^)dHj  -  dadx  -  dSdy  =  0. 

Combining  the  first  with  Equation  (2),  and  the  second 
with  (3),  we   obtain, 

-  (y  -  (3)dl3  -  {x  -  oL)doL  =  BdR,      .     (4.) 

--  dadx  —  d^dy  =  0    .     .     .     (5.) 


170  DIFFERENTIAL      CALCULUS.  [SEC.  YII, 

From  the  last  equation  we  have, 
d3  dx 

But  Equation  (2)  may  be  placed  under  the  form, 
y-^=  -^(oj-a),      or,      /3  -  y  =  -  ^(a  -  a;)  .  (V.) 

dx  d3 

Substituting  for   —  -j-  y  its  value  ~ ,  we  have, 

Since  Equations  (7)  and  (8)  are  the  same  under  dif- 
ferent forms,  they  represent  one  and  the  same  line. 

Equation  (7)  is  the  equation  of  a  normal  to  the  invo- 
lute at  a  point  whose  co-ordinates  are  x  and  y,  and  passes 
through  any  point  whose  co-ordinates  are  a.  and  (3  (Art.  44). 
Equation  (8)  is  the  equation  of  a  tangent  to  the  evolute 
at  a  point  whose  co-ordinates  are  a  and  (3,  and  passes 
through  any  point  whose  co-ordinates  are  x  and  y  (Art.  43); 
therefore, 

77ie  radius  of  curvature  which  is  normal  to  the  invo- 
lute is  tangent  to   the  evolute. 


Evolute    and    radius    of   curvatxire    increase   or  decrease    by  the 
same    quantity. 

151.     Combining  Equations  (2)  and  (G),  we  have, 

dct 

d~S 


-  =  f.(2/-^)     .    .    .    .    (9.) 


SEC.  V^II.]  INVOLUTES      AND      E  VOLUTES.  171 

Substituting  this  value   of  a;  —  a    in   Equation   ( 1 ),    we 
have,  after  reduction, 


i,-^Y(^,^)  =  ^    .   .   (10.) 


Substituting  the  same  value  in  Equation  (4),  reducing, 
and   squaring  both  members,  we  obtain, 

Dividing  (11)  by  (10),  member  by  member,  and  taking 
the  root, 

■v/<?oc2  +  d^~^  =   cm     ....     (12.) 

But  since  a  and  (3  are  the  co-ordinates  of  the  evolute, 
if  w'e  denote  this  curve  by   ^,  we  shall  have   (Art.  52), 

dB  =  dz,         dH  ~  dz  =  0,         d{Ii  -  z)   =  0; 

whence,  li  ~  z  =  a  constant  (Art.  17) ; 

which,  if  we  denote  by  c,  gives, 

li  =  z  +  c (13.) 

Since  the  difference  between  Jl  and  z  is  constant,  it  fol- 
lows that  any  change  in  one,  will  produce  a  corresponding 
and  equal  change  in  the  other. 

If  ^ve  draw  any  two  radii  of  curvature,  as  PC,  P' C\ 
jiud  denote  them  by  2i  and  Jl\  and  the  corresponding 
arcs  of  the  evolute  by   z  and   z\  we  have, 

H  =  z  +  c,        and        H'  =  z'  +  c; 

whence,  R'  ~  R  =  z'  ~  z;     that  is. 

The  difference  hetween    any   tico    radii    of  curvature   is 


172  D  T  F  F  P:  n  E  N  T  1  A  L      CALCULUS.  [SEC.  VIl. 

eq*(€d  to  fhe  arc  of  the  evolute  intercepted  between  their 
extremities. 

If  we  make  2  =  0,  and  denote  the  corresponding  value 
of  a   by   r,   we   have, 

r  =  0-{-c  =  c\    hence, 

The  constant  c,  is  equal  to  the  radius  of  curvature  passing 
through  the  origin  of  arc  of  the  evolute. 

If  we  suppose  C  to  be  the  origin  of  arc  of  the  evolute, 
then,  CP  =  r  =  c ;  and  any  radius  of  curvature,  as  C'P\ 
will  be  equal  in  length  to  the  line  C  CP.  If  then  the 
evolute  be  developed,  or  unrolled,  as  it  were,  about  the 
movable  centre  of  the  osculatory  circle,  the  other,  ex- 
tremity of  the  radius  of  curvature  will  describe  the  involute 
curve. 

Evolute    of    the    Cycloid. 

152.  Let  us  resume  the  equation  for  the  radiujs  of  curv- 
ature   (Ai't.  141), 

M  =  -  (^!_tjW    ....    (, 

dxa-y  ^    ' 

If,  in  this  equation,  we  substitute  the  value  of  dx  and 
d'^y.,  found  in  Art.  113,  we  have, 

E  =   ^'{njY  =  2^2^^.    .     .     .     (2.) 

hence  (Art.  Ill),  The  radius  of  cvrvature  is  dmdle  the 
normal ;  therefore,  when  the  generating  circle  moves  from 
^-1  towards  J/,  any  radius  of  curvature,  as  PP\  will  be 
double   the   normal    PA'. 


SEC.  VII.]  INVOLUTES      AND      EVOLUTES. 


173 


If,  in  Equation  ( 2 ),  we  make   y  =  0,    we  have,    B 
If  we  make    y  —  MB  —  2r,    we  have,    It  =  ir. 

Tliat  is,  the  radius 
of  curvature  is  zero  at 
the  point  A,  and  twice 
the  diameter  of  the 
generating  circle  at  B. 

Since  the  radius  of 
curvatlire  and  evolute 
are  both  zero  at  the 
point  A,  and  since  they 
increase    equally    (Art. 

151),  it  follows  that  the  length  of  the  evolute  AP'A'  is 
equal  to  A' JIB,  or  twice  the  diameter  of  the  generathuj 
circle. 

When  the  point  of  contact,  JST,  shall  have  reached  Jl/, 
the  poiiit  P,  will  have  described  the  involute  ABB,  and 
the  point  jP',  the  evolute  ylP'^1'.  If  we  describe  a  cii'cle 
on  A' 31  =  \A'B,  it  will  be  equal  to  the  generating  circle 
of  the  cycloid,  and  the  two  circles  Avill  touch  each  other 
at   J/.      Draw  A'X  parallel   to   AL. 

If  now  we  suppose  the  circle  whose  centre  is  C,  to  roll 
along  the  base  from  31  to  ^1,  and  the  circle  whose  centre 
IS  C",  to  roll  from  A'  to  X,  keeping  the  centres  C  and  C", 
^in  a  line  perpendicular  to  the  base  AL^  the  point  P,  of 
th.e  upper  circle,  will  re-describe  the  involute  BPA,  and 
the  point  P',  will  re-describe  the  evolute  yl'P^-1.  But 
since  the  generating  circles  are  equal,  and  since  they  are 
rolled  over  equal  bases,  the  curves  generated  will  be  equal; 
hence,   tJie  involute  and  evolute  are  equal  curves. 


174  DIFFERENTIAL      CALCULUS.  [sEC.  VH. 

Tlie  part  of  the  involute  beginning  at  A,  is  identical  with 
the  part  of  the  evolute  beginning  at   A'. 

Since  the  involute  arid  evolute  are  equal,  the  length  of 
the  involute  APB^  is  equal  to  t^vice  the  diameter  of  the 
generating  circle;  or  the  length  of  the  entire  cycloidal  arc 
APBL,  is  equal  to  the  entire  evolute  AP'A'L^  or  to- 
four  times  the  diameter  of  the  generating  circle. 

Equation  pf  the  Svolute. 

153.  The  equation  of  the  evolute  may  be  readily  found 
by  combining  the  equations, 

dx'  4-  f?y2  dyidx'  +  dy^) 

y  -  ^  =  — ^2-^'     ^-^  =  — ^^ — 1 

'\\ith  the  equation  of  the  involute  curve. 

1st.  Find,  from  the  equation  of  the  involute,  the  vaUies  of 

I        and        cr-y, 

and  substitute  them  in  the  last  two  equations ;   there  will 
result  two  new  equations,  involving    a,    /3,   jc,    and    y. 

2d.  Combine  these  equations  with  the  equation  of  the 
involute,  and  eliminate  x  and  y;  the  resulting  equation 
will  contain  a,  ^,  and  constants,  and  will  be  the  equation 
of  the  evolute  curve. 

Evolute  of  the  common  Parabola. 

154.  Let  us  take,  as  an  example,  the  common  parabola, 
of  which  the  equation  is, 

2/2  —  mx. 


SEC.  Vn.]    INVOLUTES   AND   EVOLUTES.         175 

We  shall  then  have, 

dy         m  ,„  rn^dx^ 

dx  ~  2y'  ^   ~  4y^' 

and  hence, 

and   observing  that  the  value   of  a;  —  a    is   equal  to  that 

di/ 
of  y  —  ^    multiplied  by    —  ~  •>    ^^  have, 


hence  we  have. 


4y2  _[_  ^2 
X  —  a  = ; 


rt         4^3  T  2?/2         m 

—  p    =   -^,  and         £C  —  a  =    —   -^^  —  — ; 

substituting  for  y  its  value  in  the  equation  of  the  involute, 
we  obtain, 


y  =  7n-a;% 


3 

-13   =   — ;  x-a=    -2aj--; 


and  by  eliminating  jc,   we  have, 

10/  1     \3 


P2  1^  /  1     \ 

27m\  2     /  ' 


vhich  is  the  equation  of  the  evolutc. 
If  we  make  (3  zzz  0,  we  have, 


«  =  Im; 


176  DIFFEREXTIAL      CALCULUS.  [SEC.  VD. 

an<l   hence,   the   evolute    meets  the   axis   of   abscissas   at  a 

distance    from    the    origin    equal    to    half    the    parameter. 

If  the   origin    of  co-ordinates   be 

transferred  from  A  to  this  point,  ^^V 

we  shall  have, 


1 
a'  =  a  -  -  m. 


.C' 


and  consequently, 


21m 


The  equation  of  the  curve  shows  that  it  is  symmetrical 
with  respect  to  the  axis  of  abscissas,  and  that  it  does  not 
extend  in  the  direction  of  the  negative  values  of  a'.  The 
evolute  CC  corresponds  to  the  part  ^P  of  the  involute, 
and  CC"  to  the  part  AP".  Both  are  convex  towards  the 
axis  of  -Y^ 


INTEGKAL    CALCULUS. 


Nature  of  Integration. 

155.  In  the  Differential  Calculus,  we  have  developed 
a  system  of  principles,  and  given  a  series  of  rules,  by 
means  of  which  we  deduce,  from  any  given  function,  two 
others;  the  first  of  which  is  called  the  Differential  co- 
efficient, and  the  second,  the  Differential  (Art.  25).  In 
the  Integral  Calculus,  we  have  to  return  from  the  differ- 
ential,  to  the  function  from  which   it  was   derived. 

This  operation,  as  a  fundamental  problem,  involves  the 
summation  of  a  series  of  an  infinite  number  of  terms,  each 
of  which  is  infinitely  small  (Art.  56).  No  general  rule  for 
the  summation  of  such  a  series  has  yet  been  discovered; 
and  hence,  we  are  obliged  to  resort,  in  each  particular 
case,  to  the  operation  of  reducing  the  given  differential  to 
some   equivalent   one,   whose  integral   is  known    (Art.  34). 

Forms  of  differentials  having  known  Algebraic  Functions. 

156.    We   have  found   (Art.  35),  that   every  differential 
monomial  of  the  form, 

177 


178  INTEGRAL      CALCULUS. 

in  which  m  is  any  real  number,  except  —  1,  may  be  im- 
mediately integrated ;  and  when  m  =  —  1,  the  differential 
becomes  that  of  a  logaritlunic  function,  and  its  integral 
\s  Alx  (Art.  89). 

157.    We  have  seen  that  every  differential  binomial  of 
the   form, 

(a  4-  hx'^)'^x'^-^dx^ 

in  which  the  exponent  of  the  variable  without  the  paren- 
thesis is  less  by  1  than  the  exponent  of  the  variable  with- 
in, can  be  immediately  integrated  (Art.  4i). 

15S.    We  have  seen  that  every  function  of  the  form, 

Xdx, 

in  which  X  can  be  developed  into  a  series  in  terms  of 
the  ascending  powers  of  a,  has  an  approximate  integral 
which  may  be  readily  found  (Art.  42). 


Forms   of  differentials  having  known  Logarithmic  Fonctiona. 
159.    Any  fimction  of  the  form, 

.  dx 

X 

in  which  the  numerator  is  the  differential  of  the  denom- 
inator, can  be  immediately  integrated,  since  the  integral  is 
equal  io  Alx  (Art.  89).  In  Art.  91,  we  have  given  five 
other  forms  of  differentials,  whose  corresponding  functions 
are  logarithms. 


INTEGRATION      OP      FRACTIONS.  179 

Forms    of  diflferentials  having    known  Circular    Functions. 

160.  In  Art.  99,  we  have  found  sixteen  differential 
expressions,  each  of  which  has  a  known  integi'al  corre- 
sponding to  it,  and  which,  being  differentiated,  will  of  course 
produce  the  given   differential. 

In  all  the  classes  of  functions,  any  differential  expression 
may  be  considered  as  integrated,  when  it  is  reduced  to  one 
of  the  kno\ATi  forms ;  and  the  operations  of  the  Integral 
Calculus  consist,  mainly,  in  making  such  transformations 
of  given  differential  expressions,  as  shall  reduce  them  to 
equivalent   ones,  whose   uitegrals   are   known. 


INTEGRATION     OF      RATIONAL     FRACTIONS. 

161.    Every  rational  fraction  may  be  written  under  the 
foi-m, 

Pa;«-i -f  ga;"-2    .     .     .     ^Jjx+S 
F'x^      +  Q'x^-^     .     .     .     +Ii'x+S'      ' 

in  which  the  'exponent  of  the  highest  power  of  the  variable 
in  the  numerator  is  less  by  1  than  in  the  denominator. 
For,  if  the  greatest  exponent  in  the  numerator  was  equal 
to,  or  exceeded  the  greatest  exponent  in  the  denominator, 
a  division  might  be  made,  giving  one  or  more  entire  terms 
for  a  quotient,  and  a  remainder,  in  which  the  exponent  of 
the  leading  letter  would  be  less  by  at  least  1,  than  the 
exponent  of  the  leading  letter  in  the  divisor.  The  entire 
terms  could  then  be  integrated,  and  there  would  remain 
a  fraction  under  the  above  form. 


180  INTEGRAL      CALCULUS. 

EXAMPLES. 

1.    Let  it  be  required  to  integrate  the  expression, 

ladx 

X-  —  a^ 

By   decomposing    the    denominator   into    its    factors,    we 
adx  ladx 


x^  —  a^  ~  {x  —  a)  {x  +  a) 
Let  us  make, 

2adx 


{x  —  a)  {x  ^ 


a)         \x  —  a        X  -\-  a/ 


in  which  A  and  -S  are  constants,  whose  values  may  be 
found  by  the  method  of  indeterminate  co-efficients.*  To 
find  these  constants,  reduce  the  terms  of  the  second  mem- 
ber of  the  equation  to  a  common  denominator;  we  shall 
then  have, 

adx  _  {Ax  4-  Aa  -f  ^x  —  Ba)dx 

{x  —  a)  (jc  -f  a)  "  ~~      (a;  —  a)  (a  +  a) 

Comparing  the  two  members  of  the  equation,   we  find, 
2a  =  Ax  +  Aa  +  Bx  —  Ba\ 
or,  by  arranging  with  reference  to  x, 

{A  +  B)x  +  (^  —  i?  —  2)a  — .  0;    hence, 
^  -f  i?  =  0,  and  (A  -  B  -  2)a  =  0', 

whence,  ^   =   1,         B  =   —  I. 

*  Bourdon,  Art.  11>4>     University,  Art.  1§0* 


INTEGRATION      OF      FRACTIONS.  181 

Substituting  these  values  for  A  and  JB,  we  obtain, 

2adx     _      dx  dx 

x^  —  a^  ~  X  —  a        X  +  a^ 

integrating,  we  find  (Art.  89), 

adx 


/ddx 
— =  l{x  ^  a)  —  l{x  +  a)  +  (7;   consequently, 


2.    Find  the  integral  of, 

3a;  —  5 


dx. 


aj2  —  6a;  +  8 

Resolving  the  denominator  into  its  two  binomial  factors, 
(x  —  2),   and   {x  —  4),  we  have, 

3a;  —  5  A        ,       B 

"i — a^  .   Q  =  ::: a  +  z rJ  lience. 


a;2  —  6a;  +  8        a;  - 

3a;  —  5  ^a;  —  4^  +  -i^a;  —  2^ 


2    A/y.    _L     »  ' 


3-2  __  6a;  +  8  a;-^  —  6a;  +  8 

by  comparing  the  coefficients  of  a;,   we  have, 

-  5  =   -  4^  ~  2^,         S  =  A  +  B, 

7  1 

which  gives,  -^  =  2 '        ^  ~   ~  2 ' 

Bubstitutmg  these  values,  we  have, 

f 3a  —  5  _        I  f    dx       ^    1  f     dx 

J  x^-Qx  'Vf'^  -   ~  2^  ^^^  "^  2V  ^3-4  +  ^ 


=  Jlog(a;-4)  _llog(a;-2)+  G. 


25 


•182  INTEGRAL      CALCULUS. 

Hence,  for  the  integration   of  rational  fractions: 

1st.  Hesolve  the  fraction  into  partial  fractio?is,  of  which 
the  7mmerators  shall  he  constajits^  and  the  denominators 
factors  of  the  denominator  of  the  given  fra^ction, 

2d.  Fi7id  the  values  of  the  numerators  of  the  partt(H 
fra€tio7is,  and  midtipli/  each  hy  dx. 

3d.  Integrate  each  partial  fraction  separately,  and  the 
sum  of  the  integrals  thus  found  will  be  the  integral 
sought, 

INTEGKATION      BY      PARTS. 

162.  The  integration  of  differentials  is  often  effected 
by  resolving  them  into  two  pails,  of  which  one  has  ^ 
known    integral. 

We   have   seen    (Art.  27),  that, 

d{tiv)  =z  udv  +  vduy 
whence,  by  integrating, 

uv  =  J  udv  +  /  vdu, 
and,  consequently, 

J  udv  =  uv  —Jvdu. 

Hence,  if  we  have  a  differential  of  the  form  JTdx,  which 
can  be  decomposed  into  two  factors  P  and  Qdx,  of 
which    one    of   them,    Qdx,    can    be    integrated,    we    shall 

have,  by  making   /  Qdx  =  v,   and  JP  =  u,  , 

fxdx  =fpQdx  =  J  udv  =  uv  -  Jvdu     .     (].) 
in  which  it  is  only  required  to  integrate  the  term  J  vdu. 


INTEGRATION      BY      PARTS.  183 


EXAMPLES. 


1.   Integrate  the  expression,   v?dxy/a?-  +  7?, 
This  may  be  divided  into  the  two  factors, 
£c2j         and         xdx^o?-  +  a;^, 
of  which  the  second  is  integrable  (Art.  41). 

Put,  u  =  JB^,        and        c?y  =  xdxy/d?-  +  x^\ 

then, 

.    Substituting  these  values  in  Formula  (1), 
and  finally. 


2.  Integrate  the  expression. 


(a2  -  iB2)* 


_  s 
The  two  factors  are,        aj,        and        xdx{a^  —  jb^)    ^ 

— -  1 

w  =  a; ;         c?y  =  xdx(a^  —  a;^)    - ;  v  =  - 


•/(a2  —  a2) 


/  w^y  =  ^—  +  /    , ;      whence, 

/X'^dX  X  ,  .  ,  a  ,  A  _x  N 
1  =  ^^ .  +  sm-1  -  (Art.  99). 
(a2  -  a;2)2  -/^^  -  a;2                a 


1S4:  INTEGRAL      CALCULUS, 


INTEGRATION    OF    BINOMIAL    DIFFEEENTIALS. 

t 

Form  of  BinomiaL 

163.  Every  binomial  differential  may  be  placed  imder 
the  form, 

in  "which  m  and  n  are  whole  numbers,  and  n  positive ; 
and  in  which  j9  is  entire  or  fractional,  positive  or  nega- 
tive. 

1.  For,  if   m    and  n   are  fractional,   the  binomial  takes 

the  fopm, 

i  1 

x^dx{a  +  bx^)P* 

If  we  make  x  =  2^,  that  is,  if  we  substitute  for  x,  an- 
other variable,  2,  with  an  exponent  equal  to  the  least 
common  multiple  of  the  denominators  of  the  eiqponents 
of  X,   we   shall  have, 

x^dx{a  +  bx^)P  =  ez^dz{a  +  bz^)P, 
in  which  the  exponents  of  the  variable  are  entire. 

2.  If  71  is  negative,  we  have, 

x'^-'^dx{a  +  bx-")p, 
and  by  making  a;  =  -,  we  obtain, 

-  z-'^^^dz{a  +  bz'')Py 
in  which  n  is  positive. 

3.  If   X  enters  into  both  terms  of  the  binomial,   giving 

the  form, 

x^-  dx{ax'' -\-  bx'')Py 


BINOMIAL      DIFFERENTIALS.  185 

in  which  the  lowest  power  of  x  is  -vmtten  in  the  first 
term,  we  divide  the  binomial  within  the  parenthesis  by 
ic'",  and  multiply  the  factor  without  by  cc''^;  this  gives, 

which  is  of  the  required  fomi  when  the  exponent 
*w  +  j^'-i,  is  a  whole  number,  and  may  easily  be 
reduced  to  it,   when  that   exponent  is  fractional. 


When  a  Binomial  can  be  integrated. 

164, — 1.  If  ^  is  entire  and  positive,  it  is  plain  that  the 
binomial  pan  be  integrated.  For,  when  the  binomial  is 
raised  to  the  indicated  power,  there  will  be  a  finite  nmn- 
ber  of  terms,  each  of  which,  after  being  multiplied  by 
iC'"~Wa;,    may  be  integrated   (Art.  35). 

2.  If  m  =  w,   the  binomial  can  be  integrated  (Art.  41) 

3.  K  jo  is  entire,  and  negative,  the  binomial  "will  take 
the  form, 


(a  +  hx'^y^ 
which  is  a  rational  fraction. 

•  Formula  ^. 

For  diminishing   the    exponent  of  the  v^ujabla  without  the   pa* 

renthesis. 

165.    Let  us  resume  the  difiercntw^  bmomial, 


186  INTEGRAL      CALCULTTS. 

If  we  multiply    by  the  two  factors,    jc"    and    aj-",    the 
value   will   not   be   changed,  and  we   obtain, 

Now,  the  factor  cc" -if7ic(a  +  bx'')p  is  integrable,  what- 
ever be  the  value  of  p  (Axt.  41).  Denoting  the  first 
factor,  af-"    by  w,   and  the  second  by  dv,  we  have, 

du  =   (m  —  7i)a;'"-"-Wa;,        and        v  =  -^ — .    ,,   ^    ; 
and,  consequenUy,  /a^-.&(a  +  te»).  = 

But,  /*af"  -  «  - 1  Ja;(a  +  hx'^yp  "^  -  = 

Jrffn-^-'^dx{a  +  ^ic«)?(a  +  ^"3  = 

a  f^-'^-^dxia  +  5a;")p  +  h  fx'^-'^  dx{a  +  ^"j^; 

substituting  this  last  value  in  the  preceding  equation,  and 
collecting  the  terms  containing, 

I  x"^-"^  dx{a  +  hx^y, 

af-»(a  +  5a;")P  +  i  —  a(m  —  n)  j x'^-^-'^dx{a  +  ^")^ 
"  {p  +  l)^i6  ' 


BINOMIAL      DIFFEKENTTALS.  187 

whence, 
(^) J*x'^-'^dx(a  +  bx'*)p  = 

b(2^n  +  771) 
This  formula  reduces  the  differential  binomial, 
fx'^-'^dx{a  +  bx"")?,        to         fx'^-"-'^  dx{a  +  5x")i'; 
and  by  a  similar  operation,  we  should  find, 
I ^m-n~i ^^^^ _j_  Ja;n)p^  ^q  depend  on,  / £c'"-2"-ic7ic(a  +  5af*)^ ; 

consequently,  eacA  operation  dimiiiishes  the  exponent  of 
the  variable  without  the  parenthesis  by  the  exponent  of  the 
variable  within. 

After  the  second  integration,  the  factor  m  —  n^  of  the 
second  term,  becomes  m  —  2n\  and  after  the  third, 
m  —  3w,  &c.  If  m  is  a  multij^le  of  n,  the  factor  7n  —  w, 
m  —  2/1,  m  —  3w,  &c.,  will  finally  become  equal  to  0,  and 
then  the  differential  into  Avhich  it  is  multiplied  will  disap- 
pear, and  the  given  differential  can  be  integrated.  Hence, 
a  differential  binomial  can  be  integrated^  lohen  the  ex- 
ponent of  the  variable  without  the  parenthesis  plus  1,  ia 
a  multiple  of  the  exponent  within. 

APPLICATIONS. 

166.    We  have  frequent  occasion  to  integrate  differential 
binomials  of  the  form,  - 

-y==z =  cC"  dx{a^  —  x^)    ^ . 


188 


INTEGRAL      CALCULUS. 


The  differential  binomial    x'^-'^  dx{a -{- Ix^^y    will   assume 
this  form,   if  we   substitute, 


tor 

m. 

m  +  ] 

L; 

(( 

a, 

.         a2; 

(( 

b\ 

.      -i; 

(C 

n, 

.         2; 

(C 

P^ 

.         -i. 

Making  these  substitutions  in  Formula  ^,  we  have, 


/af^dx 
y/a^  —  x^ 


ftn  —  l 


m 


ya^  -x^  -\ ^^ '- 1     .  ; 


so  that  the  given  binomial  differential  depends  on, 

/x^-'^dx 
'^a?  —  ^ 

and  in  a  similar  manner  this  is  found  to  depend  upon, 

x"^-^  dx 


and  so  on,  each  operation  diminishing  the  exponent  of  a?  by 
2.    If  m  is  an  even  number,  the  integral  will  depend,  after 


m 


operations,   on   that   of. 


/--—^^ ,,,  sin-i  -    (Art.  99). 

J  Va"  -  x^  « 


binomial    dipfeeentials.  189 

Formula  2S. 
For  diminishing  the  exponent  of  the  parenthesis. 

167.    By   changing    the    form  of   the    given   differential 
binomial,  we  have, 

fx'^-'^dx(a  +  hx'')P-'^  (a  +  bx"")  = 
a  fx'^-'^  dx{a  +  6a;")?-i  +  b  fx'^+''-'^dx{a  +  5a;")i'-i. 

Applying  Formula  ^  to  the  second  term,  and  observing 
that  m  is  changed  to  m  +  w,  and  ^  to  j9  —  1,  we  have, 

af(a  4-  bx'^)P  —  am  / xf^-^  dx{a  +  Ja;")^-^ 
^(/??i  +  m) 

Substitutmg  this  value  in  the  last  equation,  we  have, 
(IB) Jx"^-^ dx{a  +  bx'')P  z= 

af»(a  +  bx'^y  -h  pna  I  x"^-"^  dx{a  +  bx")?-"^ 
pn  +  m  ' 

in   which  the  exponent  of  the  parenthesis  is  diminished  by 
],  for  each  operation. 


APPLICATIONS. 

1 .    Integrate  the  expression    dxla^  -f  x^Y  . 

The   differential  binomial    x'^-'^dx{f:i  +  bx'')p  will  assume 


190  INTEGRAL      CALCULUS. 

this    form,    if   we    make    m  =  1,    a  =  d^^    6  =  1,    n  =  2, 
and   jt>  =  f . 

Substituting  these  values  in  the  formula,  we  have, 

jdx{a?'  +  x'Y  =  -^ 

Applying  the  formula  a  second  time,  we  have, 


7? 


But  we  have  found   (Art.  9l), 

*^  yd'  +  x^ 
^^"^^'  .  /(/^.(a^  +  x^)i  = 

— — ^ — '—  +  ^dx^ — ^  +  —  'l\x  ^  a/«   +  «V  +  ^^ 


2.    Integrate  the   expression,    dx^^r^  —  x^. 

The  first  member  of  the  equation  will  assume  this  form, 
if  we  make,  m  =  1,  a  =  r^,  6  =  —  1,  w  =  2,  and 
jt)  =  i.  Substituting  these  values  in  the  formula,  we 
have, 

whence,  by  substitution   (Art.  99), 


^^J^ 


binomial    diffeeentials.  191 

Formula    <S. 

For  diminishing   the  exponent    of  the  variable  without    the 
parenthesis,   when  it  is  negative. 

168.  It  is  evident  that  Formula  ^  will  only  diminish 
m  —  1,  the  exponent  of  the  variable,  when  m  is  positive. 
We  are  now  to  determine  a  formula  for  diminishing  this 
exponent  when  m  is  negative. 

From  Formula  A,  we  deduce, 

a{m  —  n)  ' 

changing  m,  to   —  m  +  w,   we  have, 

(^) Jx-'^-^dx{a  +  ^aj")P  = 

«-'"(«  +  bx'')p-^^  +  5(m  —  w  —  n]))  fx-"'+"-'^dx{a  +  ^ic«)i' 

—  a  in  ' 

in  which  formula,  it  should  be  remembered  that  the  nega- 
tive sign  has  been  attributed  to  the  exponent  m. 


APPLICATIONS. 


1.   Integrate  the  expression    x-^dx{2  —  x"^ 

The  first  member  of  Equation  {'^)  will  assume  this  form, 
if  we  make  'm  =  I,  a  =  2^  b  =  —  1,  7i  =  2,  and 
JO  =   —  f .      Substituting  these   values,  we   have, 

_i 
fx-^dx{2  -  x-'f^  =z   -  ^"'(2  -  a^^)       _^y^2  _  x^f'dx. 


102  INTEGRAL      CALCULUS. 

The  (lifTerential  term  iii  the  second  member  "will  be  in- 
tegrated by  the  next  formula. 

Formula  22>. 

For    diminishicg    the    exponent    of   the    parenthesis    when    it    is 

negative. 

169.  It  is  evident  that  Formula  ^I^  will  only  diminish 
p,  the  exponent  of  the  parenthesis,  when  p  is  positive. 
We  are  now  to  determine  a  formula  for  diminishing  this 
exponent  when  p  is  negative. 

We  find,  fi-om  Formula  2S, 

pna  ' 

T\Titing  for   2h    —  i^  +  Ij    "^^  have, 

(2>) fx^-hJx(a  +  hx^)-P  ^ 

x'"{a.  +  bx'')-P+^  —  (m  +  ;i  —  ;?;3)  fx'^-hlxia  +  5a;«)-i'-»-i 
7ia{2)  —  I) 

When  p  z=z  1,  ^>  —  1  =z  0 ;  the  second  member  be- 
comes infinite,  and  the  given  expression  becomes  a  ra- 
tional fraction. 

APPUCATTOXS. 

1.  Integrate  the  expression,    /  dx{2  —  x^)    ^. 


^ 


BINOMIAL      DIFFE 

The  first  member  of  Equation  ^ 
if  we  make  m  =  1,  a  =  2,  I 
pz=  —^,    Substituting  these  values, 

since  the   coefficient   of  the   second  term,   in    the  formrJa, 
becomes  zero. 

Returning,  then,  to  the  example  under  the  last  formula, 
we  have, 

J  x-^dx{2  -x^)    "  = ^—- '—  +  -^—Y'  ^' 

2.  By  means  of  Formula  3J>,  we  are  able  to  mtegrate 
the  expression, 

(^-^  =  <^^-'  +  ^^)-''. 

when  p  is  a  whole  number. 

The  general  formula  will  assume  this  foim,  if  we  make 
m  =  1,    X  =  z,     a  =  a"^,     b  =  1,    n  =  2. 

Each  application  of  the  formula  will  reduce  the  expo- 
nent —  p,  by  1,  until  the  integral  will  finally  depend  on 
that  of 

_^^  =  itan-^-  +  G    (Art.  99). 

a^  +  z^        a  a  ' 


Formula  2^. 

When  the  variable  enters  into   both  terms  of  the  binomiaL 

lyo.    Let  it  be  required  to  integrate  the  expression, 

xidx  ^  ,  „.-4 

=  xidx{2ax  —  x^)    ^. 


'yjlax  — 


AL      CALCULUS. 

nay  be  placed  under  the  form, 

1  .  -^'     ^)   t)y  making, 

we  shall  then  have, 

fx~^dx{2a  -  x)~^  = 

X     "(2a  -  «)^    ,    2a(q  -  ^)  p  f-f  ,  ,„  ,-^ 
V L_  _| \i J  ^     ^dx(2a  —  x)  ^ 


1 
If  we  observe  that, 


jc     ^  =  a;      a",        and        x     ^  =  x     x  ^, 

and  pass  the  fractional  powers  of  x  within  the  parentheses, 

we  shall  have, 

x^dx 


(^) /-7. 


\/2ax  —  x^ 


x9-'^y2ax  —  jb2  ^  (2q  —  l)a  P    xi-^dx 


+ 


9.  9.        "^  ^/2ax  —  x^ 

Each  application  of  this  formula  diminishes  the  expo- 
nent of  the  variable  without  the  parenthesis  by  1.  If  3' 
is  a  positive  and  entire  nimiber,  we  shall  have,  after  q 
reductions, 

f—J^=r=  =  ver-sin-^  -  +  <7    (Art.  99). 
*^  ^/2ax  -  a;2  « 


•tr^  jr-v:  .      '  J^T/'  ''^\^'r,, 


.,y  ■  - 


A 


'.^■^TfVM 


v9!fW'r,'W1f^(fr.r-rrjiffjni(pxr'.,  ■?»» 


